GROUPED PATTERNS OF HETEROGENEITY IN PANEL DATA Stéphane Bonhomme and Elena Manresa CEMFI Working Paper No. 1208 June 2012 CEMFI Casado del Alisal 5; 28014 Madrid Tel. (34) 914 290 551 Fax (34) 914 291 056 Internet: www.cemfi.es We thank Daron Acemoglu, Daniel Aloise, Dante Amengual, Manuel Arellano, Jushan Bai, Alan Bester, Fabio Canova, David Dorn, Kirill Evdokimov, Ivan Fernandez-Val, Lars Hansen, Jim Heckman, Han Hong, Bo Honoré, Andrea Ichino, Jacques Mairesse, Serena Ng, Taisuke Otsu, Eleonora Patacchini, Peter Phillips, Enrique Sentana, Martin Weidner, and seminar participants at CAM, CEMFI, Columbia University, CREST, EUI, IFS, Princeton University, Toulouse School of Economics, UCL, University of Chicago, and Yale University for useful comments. Support from the European Research Council/ ERC grant agreement nº 263107 is gratefully acknowledged. All errors are our own.
We thank Daron Acemoglu, Daniel Aloise, Dante Amengual, Manuel Arellano, Jushan Bai, Alan Bester, Fabio Canova, David Dorn, Kirill Evdokimov, Ivan Fernandez-Val, Lars Hansen, Jim Heckman, Han Hong, Bo Honoré, Andrea Ichino, Jacques Mairesse, Serena Ng, Taisuke Otsu, Eleonora Patacchini, Peter Phillips, Enrique Sentana, Martin Weidner, and seminar participants at CAM, CEMFI, Columbia University, CREST, EUI, IFS, Princeton University, Toulouse School of Economics, UCL, University of Chicago, and Yale University for useful comments. Support from the European Research Council/ ERC grant agreement nº 263107 is gratefully acknowledged. All errors are our own.
CEMFI Working Paper 1208 June 2012
GROUPED PATTERNS OF HETEROGENEITY IN PANEL DATA
Abstract This paper introduces time-varying grouped patterns of heterogeneity in linear panel data models. A distinctive feature of our approach is that group membership is left unspecified. We estimate the model’s parameters using a “grouped fixed-effects” estimator that minimizes a least-squares criterion with respect to all possible groupings of the cross-sectional units. We rely on recent advances in the clustering literature for fast and efficient computation. Our estimator is higher-order unbiased as both dimensions of the panel tend to infinity, under conditions that we characterize. As a result, inference is not affected by the fact that group membership is estimated. We apply our approach to study the link between income and democracy across countries, while allowing for grouped patterns of unobserved heterogeneity. The results shed new light on the evolution of political and economic outcomes of countries. Keywords: Discrete heterogeneity, panel data, fixed effects, democracy. JEL Codes: C23. Stéphane Bonhomme CEMFI [email protected]
Unobserved heterogeneity is central to applied economics. There is ample evidence that workers and
firms differ in many dimensions that are unobservable to the econometrician (Heckman, 2001). Cross-
country analyses also show evidence of considerable heterogeneity (e.g., Durlauf et al., 2001). In view
of this prevalence, the use of flexible empirical approaches to model unobserved heterogeneity has been
advocated in the literature (e.g., Browning and Carro, 2007). In practice, however, there is a trade-off
between specifying rich patterns of heterogeneity, and building parsimonious specifications that are
well adapted to the data at hand. The goal of this paper is to propose a flexible yet parsimonious
approach to deal with the presence of unobserved heterogeneity in a panel data context.
A widely used approach in applied work is to model heterogeneous features as unit-specific, time-
invariant fixed-effects. Fixed-effects approaches (FE) are conceptually attractive, as they allow for
unrestricted correlation between unobserved effects and covariates. When one is interested in mea-
suring the effect of one particular covariate, this means that general fixed-effects endogeneity is taken
care of in estimation.
However, allowing for as many parameters as individual units comes at a cost. Lack of parsimony
implies that estimates of common parameters are subject to an “incidental parameter” bias that may
be substantial in finite samples (Nickel, 1981, Hahn and Newey, 2004). Moreover, the unit-specific
fixed-effects are typically poorly estimated in short panels, often preventing the researcher to make
sense of unobserved heterogeneity estimates.1 In addition, FE may not be that flexible either. Indeed,
although standard time-invariant FE approaches model cross-sectional heterogeneity in a flexible way,
they are severely restricted in the time-series dimension.
This paper proposes a different approach to model unobserved heterogeneity, which has three main
features. First, unlike standard FE, we allow heterogeneity patterns to vary over time in a flexible
manner. Second, our modelling strategy relies on the assumption that time patterns are common
within groups of individuals. Finally, our approach shares with FE the property that it leaves the
relationship between observables and unobservables unrestricted, thus allowing for general forms of
covariates endogeneity.
A simple linear model with grouped patterns of heterogeneity takes the following form:
yit = x′itθ + αgit + vit, i = 1, ..., N, t = 1, ..., T, (1)
where the covariates xit are contemporaneously uncorrelated with vit, but may be arbitrarily corre-
lated with the group-specific unobservables αgit. The group membership variables gi ∈ 1, ..., G are
1As an example, estimation of the fixed effects has received some attention in the literature on school and teacher
quality (Kane and Staiger, 2002). In short panels, it may be possible to consistently estimate features of the cross-sectional
distribution of the individual fixed-effects, as opposed to the individual effects themselves (Arellano and Bonhomme,
2011).
2
unrestricted, and will be estimated along with the other parameters of the model. The group-specific
time dummies αgt are fully unrestricted as well. Lastly, the number of groups G is to be set or esti-
mated by the researcher. The baseline framework of model (1) may easily be modified to incorporate
restrictions on the group-specific time patterns, and to allow for additive time-invariant fixed-effects
in addition to the time-varying grouped effects.
There are theoretical and empirical reasons for considering group-specific patterns of heterogeneity.
As a first example, the static group interaction model for panel data (e.g., Blume et al., 2010) may
be seen as a special case of model (1), where αgit includes means of covariates and outcomes. In
this context, our framework may be used to estimate the reference groups, simply by treating αgit
as unrestricted parameters. As a second example, tests of full risk-sharing in village economies are
also often based on the same model (Townsend, 1994, Munshi and Rosensweig, 2009).2 Note that, in
contrast with most applications of social interactions and risk sharing models, our approach allows to
estimate the reference groups from the data, under the assumption that group membership remains
constant over time.
In many empirical applications, interdependence across units is treated as a nuisance, and taken
into account using robust (“clustered”) standard errors formulas. Yet, dependence per se may often be
of interest to the researcher. In this perspective, grouped patterns of heterogeneity can be interpreted
as a flexible way of modelling interdependence across individual units over time. Compared to existing
spatial dependence models for panel data (e.g., Sarafidis and Wansbeek, 2012), model (1) allows the
researcher to estimate the spatial weights matrix. This relaxes an important requirement of these
models, where the notion of “economic distance” is sometimes elusive.
A distinctive feature of our approach is that group membership is estimated from the data. Our
estimator is based on an optimal grouping of the N units, according to a least-squares criterion. This
approach is statistically well grounded, as it delivers a consistent estimator of the model’s parameters
under correct specification, and as it allows the researcher to compute standard errors that take into
account the fact that groups have been estimated. Nonetheless, in contexts where the researcher has
some a priori information on group composition, our estimator may easily be extended to incorporate
this information to the estimation problem in a non-dogmatic way.
Note that the group membership variables gi may be viewed as indexing the N time-varying
sequences of unit-specific unobserved heterogeneity. The key assumption is that at most G of these
sequences are distinct from each other. This restricts the support of the unobserved heterogeneity,
while leaving other features of the relationship between observables and unobservables unrestricted.3
2In tests of full insurance, yit in model (1) would be (the first difference of) log household consumption. An important
assumption for the test to be valid is that households have common (CRRA) preferences, see for example Shulhofer-Wohl
(2011).3In this sense, our approach is reminiscent of sparsity assumptions that have been widely studied in regression models
(Tibshirani, 1996).
3
Thus, our treatment of grouped heterogeneity differs from finite mixture models, since these models
rely on assumptions that restrict the relationship between unobserved heterogeneity and observed
covariates.4 In contrast, and in close analogy with fixed-effects, our approach leaves that relationship
unspecified.
Our estimator, which we will refer to as “grouped fixed-effects” (GFE), relies on an optimal group-
ing of the cross-sectional units. Determining the optimal grouping represents a computational chal-
lenge. Fortunately, this problem has been extensively studied by the research community working on
data clustering (Steinley, 2006). In the absence of covariates in model (1), the estimation problem
coincides with the standard minimum sum-of-squares partitioning problem, and a simple solution is
given by the “kmeans” algorithm (Forgy, 1965). Making use of the connection with the clustering
literature, we compute the GFE estimator using a state-of-the-art heuristic approach (Hansen et al.,
2010), which we extend to allow for covariates.
Our algorithm delivers a fast and reliable solution to the computation problem.5 We assess its
performance by building on recently proposed exact solution algorithms (Brusco, 2006, Aloise et
al., 2009). The numerical experiments that we have performed suggest that our algorithm correctly
identifies the globally optimal grouping, at least in datasets of moderate size such as the one that
we use in our empirical application. This encouraging evidence confirms previous results obtained for
minimum sum-of-squares partitioning (Brusco and Steinley, 2007).
We derive the properties of the grouped fixed-effects estimator in an asymptotic where N (the
number of units) and T (the number of time periods) tend to infinity simultaneously.6 Although the
estimator is biased for small T , the bias vanishes at a faster-than-polynomial rate provided groups are
well separated, and errors vit satisfy suitable tail and dependence conditions. Under these assumptions,
the GFE estimator is automatically (higher-order) bias-reducing, and it is asymptotically equivalent
to the infeasible least squares estimator in which the population groups are known. This finding has
implications for applied work, as standard errors are unaffected by the fact that the group membership
variables have been estimated.
The asymptotic properties of our estimator contrast with available results for models with unit-
specific fixed effects, where the incidental parameter bias is of the O(1/T ) order in general (Arellano
and Hahn, 2007). At the heart of the difference is the fact that group classification improves very fast
4See the monographs by McLachlan and Peel (2000) and Fruhwirth-Schnatter (2006) for recent advances in this area.
Important contributions in economics include Heckman and Singer (1984) and Keane and Wolpin (1997). Kasahara and
Shimotsu (2009) and Browning and Carro (2011) study identification in finite mixtures of discrete choice models for a
fixed number of groups. Geweke and Keane (2007) and Norets (2010) are recent examples of flexible modelling strategies.5Executable codes (coded in FORTRAN), as well as a Stata replication of the empirical results, are available at:
http://www.cemfi.es/∼bonhomme/6Previous results obtained for the minimum sum-of-squares partitioning problem (Pollard, 1981, 1982) were derived
in an asymptotic where T is kept fixed as N tends to infinity. In this setting, parameter estimates are inconsistent in
as the number of time periods increases. A related result was recently obtained by Hahn and Moon
(2010) in a class of nonlinear models with discrete time-invariant heterogeneity. Relative to Hahn and
Moon, this paper allows for time-varying heterogeneity and provides primitive conditions in the case
of the linear model. Interestingly, we also find that adding non-dogmatic prior information to GFE
does not affect the large-T properties of the estimator, in sharp contrast with fixed-effects models
(Arellano and Bonhomme, 2009).
The grouped fixed-effects estimator is also related to factor-analytic, “interactive fixed-effects”
estimators (Bai, 2009). Indeed, the GFE model of unobserved heterogeneity has a factor-analytic
structure, as:
αgit = (α1,t, α2t, ..., αGt)︸ ︷︷ ︸f ′
t
× (0, 0, ..., 1, ..., 0)′︸ ︷︷ ︸λi
.
Unlike interactive FE, which recover the structure of heterogeneity up to an unknown rotation, the
GFE approach recovers the exact group structure. In addition, for a given number of groups the GFE
approach is more parsimonious than factor-analytic ones, resulting in smaller asymptotic biases under
correct specification. This parsimony may be useful in situations where the data are not informative
enough to allow for fully unrestricted interactive effects.
We take advantage of the mathematical connection with interactive fixed-effects models to conduct
the asymptotic analysis. In particular, we use an insight from Bai (1994, 2009) to establish consistency
of the GFE estimator. We also rely on the analysis of Moon and Weidner (2010b) to discuss the
important issue of misspecification of the number of groups. As an example, we show that estimating
two groups when the data generating process is homogeneous does not bias the slope estimator.
However, the bias on the intercept(s) can be substantial. Lastly, we rely on Bai and Ng (2002) to
propose a class of information criteria that consistently select the true number of groups as N and T
tend to infinity.
We use our approach to study the link between income and democracy on a panel of countries that
spans the last part of the twentieth century. In an influential paper, Acemoglu et al. (2008) find that
the well-documented positive association between income and democracy disappears when controlling
for additive country- and time-effects in a panel dataset. They interpret the country-specific fixed-
effects as reflecting long-run, historical factors that have shaped political and economic development
of countries. However, FE may not be the most appropriate method on these data: the within-country
variance of income is small, and the estimates of country-specific heterogeneity are very imprecise due
to the short length of the panel– seven five-year periods in our benchmark dataset. In addition, FE
ex-ante rules out time-varying patterns of heterogeneity, in a period that is characterized by a large
number of transitions to democracy.
We revisit the evidence using the grouped fixed-effects approach. Our benchmark results are based
on model (1), which allows for time-varying grouped patterns of unobserved country heterogeneity.
This modelling is consistent with the empirical observation that regime types and transitions tend to
5
cluster in time and space, as documented in the political science literature (e.g., Gleditsch and Ward,
2006, Ahlquist and Wibbels, 2012). An early conceptual framework is laid out in Huntington (1991)’s
work on the “third wave of democracy”, which argues that international and regional factors– such as
the influence of the Catholic Church or the European Union– may have induced grouped patterns of
democratization.
According to the baseline specification, the effect of income on democracy remains positive and
significant when allowing for grouped patterns of heterogeneity. This effect is quantitatively small,
as a result of a substantial endogeneity bias in the cross-section. Moreover, the income effect disap-
pears when allowing for time-varying grouped effects and time-invariant country-specific fixed-effects
simultaneously.
Our main empirical finding is that estimates of the time-varying country-specific determinants of
democracy are not consistent with an additive fixed-effects specification. Specifically, while approx-
imately two thirds of the countries in our sample display stable time profiles over the period, one
third of the sample experiences a clear upward trend. Two of the groups that we identify comprise
transition countries– in Southern Europe and Latin America, and in part of Africa– whose democracy
levels show substantial increases at different points in time.
An important question is then why the estimated time profiles differ across countries. To explore
this issue, we regress the estimated groups on various factors that the literature has pointed out
as potential determinants of democracy. As a particular historical, long-run determinant, we use a
measure of constraints on the executive at the time of independence constructed by Acemoglu et al.
(2005, 2008). We find that constraints at independence were significantly more stringent in countries
that remained democratic between 1970 and 2000, compared to those that remained non-democratic.
However, this measure does not explain why some countries that were non-democratic at the beginning
of the sample period experienced a democratic transition, while others did not. These results call for
further study of the short- and long-run determinants of democracy. For a sizable share of the world,
history appears to have evolved at a fast pace.
To end this introduction, note that this paper is not the first one to rely on group structures
for modelling unobserved heterogeneity in panels. Bester and Hansen (2010) show that grouping
individual fixed-effects may result in gains in precision. In their setup, heterogeneity is time-invariant
and the grouping of the data is assumed known to the researcher. A recent paper by Lin and Ng
(2011) considers a random coefficients model and uses the time-series regression estimates to classify
individual units into several groups. They also propose a classification algorithm that is related to
ours, although they do not derive the asymptotic properties of the corresponding estimator. None of
these two papers allows for time-varying unobserved heterogeneity.7
7Group models and clustering approaches have also been used to search for “convergence clubs” in the empirical growth
literature; see for example Canova (2004), and Phillips and Sul (2007). Yet another related work is Sun (2005), who
considers parametric finite mixture models for panel data and studies the properties of maximum likelihood estimation.
6
The outline of the paper is as follows. In Section 2 we introduce the grouped fixed-effects estimator
and several extensions. In Section 3 we discuss computation issues. In Section 4 we derive the
asymptotic properties of the estimator as N and T tend to infinity. Section 5 considers inference and
estimation of the number of groups, and provides some finite sample evidence on the performance of
the estimator. In Section 6 we use the GFE approach to study the relationship between income and
democracy. Lastly, Section 7 concludes.
2 The grouped fixed-effects estimator
We start by introducing the grouped fixed-effects (GFE) estimator in the baseline model (1). Then
we outline several extensions.
2.1 Baseline model
Model (1) contains three types of parameters: the parameter vector θ ∈ Θ, which is common across
individual units; the group-specific time dummies αgt ∈ A, for all g ∈ 1, ..., G and all t ∈ 1, ..., T;and the group membership variables gi, for all i ∈ 1, ..., N, which map individual units into groups.
The parameter spaces Θ and A are subsets of RK and R, respectively. We denote as α the set of all
αgt’s, and as γ the set of all gi’s. Thus, γ ∈ ΓG denotes a particular grouping of the N units, where
ΓG is the set of all groupings of 1, ..., N into (at most) G groups.
It is assumed that xit and vit are weakly uncorrelated. In particular, the covariates vector xit
may include strictly exogenous regressors, lagged outcomes, or general predetermined regressors. The
model also allows for time-invariant regressors under certain support conditions. In contrast, xit
and αgit are allowed to be arbitrarily correlated. We defer a more precise statement of the required
assumptions until Section 4.
The grouped fixed-effects estimator is defined as the solution to the following minimization problem:
(θ, α, γ
)= argmin
(θ,α,γ)∈Θ×ANT×ΓG
N∑
i=1
T∑
t=1
(yit − x′itθ − αgit
)2, (2)
where the minimum is taken over all possible groupings γ = g1, ..., gN of the N units into G groups,
common parameters θ, and group-specific time effects α.
For computational purposes, as well as to derive asymptotic properties, it is convenient to intro-
duce an alternative characterization of the GFE estimator based on concentrated group membership
variables. It is easy to see that, for any given values of θ and α, the optimal assignment for each
individual unit is:
gi (θ, α) = argming∈1,...,G
T∑
t=1
(yit − x′itθ − αgt
)2, (3)
where we take the minimum g in case of a non-unique solution.
7
The GFE estimator of (θ, α) in (2) is then equivalently written as:
(θ, α
)= argmin
(θ,α)∈Θ×AGT
N∑
i=1
T∑
t=1
(yit − x′itθ − αgi(θ,α)t
)2, (4)
where gi (θ, α) is given by (3). The GFE estimate of gi is then simply gi
(θ, α
).
Two remarks are in order. First, unlike standard finite mixture modelling (McLachlan and Peel,
2000), where the group probabilities are specified as parametric or semiparametric functions of ob-
served covariates, the grouped fixed-effects approach leaves group probabilities unrestricted. In fact, we
show in Appendix B that the GFE estimator maximizes the pseudo-likelihood of a mixture-of-normals
model, where the mixing probabilities are unrestricted and individual-specific. In this perspective,
the grouped fixed-effects approach may be viewed as a point of contact between finite mixtures and
fixed-effects.
Secondly, one can see, from (4), that the grouped fixed-effects estimator minimizes a piecewise-
quadratic function where the partition of the parameter space is defined by the different values of
gi (θ, α), for i = 1, ..., N . On each element of this partition, the GFE objective is a simple quadratic
function, corresponding to the least squares objective in the regression of yit on xit and interactions
of group and time dummies. The criterion function is thus non-standard: although it is globally
continuous, it is neither globally differentiable nor convex as soon as G > 1. Moreover, the number
of partitions of N units into G groups increases steeply with N , making exhaustive search virtually
impossible. As a result of its complexity, the GFE objective may have a large number of local minima.
In the next section we will rely on recent advances in the literature on data clustering to address this
computational difficulty.
2.2 Extensions
Here we outline several simple extensions of the baseline grouped fixed-effects model that may be
useful for applied work. We end the section by briefly describing a general GFE estimator for nonlinear
models.
Unit-specific heterogeneity. One simple generalization is to allow for both time-invariant fixed
effects and time-varying grouped effects as follows:
yit = x′itθ + αgit + ηi + vit, (5)
where ηi are N unrestricted parameters. Denoting unit-specific means as wi =1T
∑Tt=1wit, (5) yields
the following equation in mean deviations:
yit − yi︸ ︷︷ ︸yit
= (xit − xi)︸ ︷︷ ︸xit
′θ + αgit − αgi︸ ︷︷ ︸αgit
+ vit − vi︸ ︷︷ ︸vit
, (6)
8
which may be estimated using grouped fixed-effects.8
Modelling time patterns. Another simple extension is to impose linear constraints on the group-
specific time effects αgt, as in: αgt =∑R
r=1 α(r)g ψr (t) where ψ1, ..., ψR are known functions, and α
(r)g
are scalar parameters to be estimated. Linear constraints are easy to embed within the computational
and statistical framework of model (1), and allow to model a wide variety of patterns of unobserved
heterogeneity.
As an example, in the empirical application we show estimates of a model with two different layers
of heterogeneity that takes the following form:
yit = x′itθ + αgi1t + ηgi1,gi2 + vit, (7)
where (gi1, gi2) ∈ 1, ..., G1 × 1, ..., G2 indicates joint group membership. This model may be
interpreted as a restricted version of model (1) with G = G1×G2 groups, and with G1(G2− 1)(T − 1)
linear constraints on the group-specific time dummies.9
Adding prior information. In certain applications researchers may want to use prior information
on the structure of unobserved heterogeneity. For example, in a cross-country application one could
think that countries in the same continent share some dimensions that are unobserved to the econome-
trician. In such situations, one possibility is to impose the group structure on the data by assumption,
e.g. by controlling for continent dummies possibly interacted with time effects. Another possibility
is to use our grouped fixed-effects estimator, which leaves the groups unrestricted and recovers them
endogenously. An intermediate possibility is to combine a priori information on group membership
with data information, simply by adding a penalty term to the right-hand side of (2). See Appendix
B for details on this alternative approach.
Nonlinear models. To conclude this section, we note that the GFE approach may be applied to
nonlinear models also. A general M-estimator formulation based on a data-dependent function mit(·)is as follows:
(θ, α, γ
)= argmin
(θ,α,γ)∈Θ×ANT×ΓG
N∑
i=1
T∑
t=1
mit (θ, αgit) . (8)
8Our asymptotic results imply that GFE yields large-T consistent estimates of the model’s parameters in (6) if covari-
ates are strictly exogenous or predetermined. In contrast, GFE is generally inconsistent in the presence of endogenous
covariates– which do not satisfy E(xitvit) = 0. In this case, (GFE analogs of) instrumental variables strategies are
required.9Let µg1g2t
= αg1t + ηg1,g2. It is easy to see that the following G1(G2 − 1)(T − 1) linear constraints are satisfied:
µg1g2t−
1
T
T∑
s=1
µg1g2s−
1
G2
G2∑
h=1
µg1ht+
1
G2T
G2∑
h=1
T∑
s=1
µg1hs= 0, for all (g1, g2, t) .
9
This framework covers random coefficients models and likelihood models as special cases.10 In partic-
ular, it encompasses static and dynamic discrete choice models. However, studying the properties of
GFE in nonlinear models raises a number of challenges, which we do not address in this paper.
3 Computation
Computation of the grouped-fixed effects estimator is particularly challenging due to the piecewise-
quadratic nature of the criterion. Given its accused non-convexity, and the large number of local
minima, direct minimization is not well-suited. As an alternative, we exploit a connection with data
clustering and take advantage of recent developments in this literature in order to obtain fast and
efficient computation methods.
3.1 Algorithms
We present two computation algorithms in turn: a simple iterative scheme, and a more efficient
alternative.
A simple iterative algorithm. A simple strategy to minimize (4) is to iterate back and forth
between group classification (computation of gi) and estimation of the common parameters (θ and α),
until numerical convergence. This may be done as in the following iterative algorithm.
Algorithm 1 (iterative)
1. Let(θ(0), α(0)
)∈ Θ×AGT be some starting value.
Set s = 0.
2. Compute for all i ∈ 1, ..., N:
g(s+1)i = argmin
g∈1,...,G
T∑
t=1
(yit − x′itθ
(s) − α(s)gt
)2. (9)
3. Compute:(θ(s+1), α(s+1)
)= argmin
(θ,α)∈Θ×AGT
N∑
i=1
N∑
t=1
(yit − x′itθ − α
g(s+1)i t
)2. (10)
4. Set s = s+ 1 and go to Step 2.
10A GFE estimator in the random coefficients model is obtained by taking mit (αgit) = (yit − x′
itαgit)2. Note that
in this case A is a subset of RK , where K = dimxit. A GFE estimator in a likelihood setup is obtained by taking
mit (θ, αgit) = − ln f (yit|xit; θ, αgit), where f(·) denotes a parametric density function.
10
Algorithm 1 alternates between two steps. In the “assignment” step, each individual unit i is
assigned to the group g(s+1)i whose vector of time effects is closest (in an Euclidean sense) to her
vector of residuals. In the “update” step, θ and α are computed given the group assignment. Note
that (10) corresponds to a simple OLS regression that controls for interactions of group indicators and
time dummies.11
This simple iterative scheme is a clustering algorithm. Indeed, it coincides with the well-known
kmeans algorithm (Forgy, 1965) in the special case where there are no covariates in the model (i.e.,
when θ = 0).12 In this case, (4) boils down to the standard minimum sum-of-squares partitioning
problem:
α = argminα∈AGT
N∑
i=1
(min
g∈1,...,G
T∑
t=1
(yit − αgt)2
). (11)
In geometric terms, (11) amounts to finding a collection of “centers” α1, α2, ..., αG in RT such
that the sum of the Euclidean distances between yi and the closest center αg is minimum. Due to its
relevance in many different fields (such as astronomy, genetics or psychology), this problem has been
extensively studied in operations research and computer science (Steinley, 2006). We build on recent
advances in that literature in order to develop an efficient extension of Algorithm 1, and to construct
computation-intensive exact algorithms that serve to assess its performance.
It is easy to see that, in Algorithm 1, the objective function on the right-hand side of (4) is non-
increasing in the number of iterations. Numerical convergence is typically very fast. However, a major
drawback of Algorithm 1 is its dependence on the chosen starting values. A simple way to overcome
this problem is to choose many random starting values, and then select the solution that yields the
lowest objective. In the application we will use the following method to generate starting values:13
1. Draw θ(0) from some prespecified distribution supported on Θ.
2. Draw G units i1, i2, ..., iG in 1, ..., N at random, and set:
α(0)gt = yigt − x′igtθ
(0), for all (g, t).
11As written, the solution of the algorithm may have empty groups. A simple modification consists in re-assigning one
individual unit to every empty group, as in Hansen and Mladenovic (2001). Note that doing so automatically decreases
the objective function.12Note that similar iterative schemes will apply to more general (possibly nonlinear) models. See for example the
literature on “clusterwise regression” in operations research (Spath, 1979, Caporossi and Hansen, 2005), and more
recently Lin and Ng (2011).13See Maitra, Peterson and Ghosh (2011) for a comparison of various initialization methods for the kmeans algorithm.
Another simple initialization scheme that we have considered it to select G+ r units at random, and to set(θ(0), α(0)
)
as the global minimum of the GFE objective in that subsample. This can be done easily for low values of r. A practical
advantage of this method is that the researcher does not need to prespecify a distribution for θ(0). In our experiments,
we observed very little difference between the two initialization methods.
11
A more efficient algorithm. In practice, as in kmeans, a prohibitive number of starting values
may be needed to obtain reliable solutions. The Variable Neighborhood Search (VNS) method has
been pointed out as the state-of-the-art heuristics to solve the minimum sum-of-squares partitioning
problem (Hansen and Mladenovic, 2001, Hansen et al., 2010). We extend the specific algorithm used
in Pacheco and Valencia (2003) and Brusco and Steinley (2007) to allow for covariates. The algorithm
works as follows, where as before γ = g1, ..., gN is a generic notation for a grouping of the N units
into G groups.
Algorithm 2 (Variable Neighborhood Search)
1. Let (θ, α) ∈ Θ×AGT be some starting value.
Perform one assignment step of Algorithm 1 and obtain an initial grouping γ.
Set timemax and nmax to some desired values.
Set γ∗ = γ.
2. Set n to 1.
3. Relocate n randomly selected units to n randomly selected groups, and obtain a new grouping γ′
.
Perform one update step of Algorithm 1 and obtain new parameter values(θ′
, α′
).
4. Set(θ(0), α(0)
)=(θ′
, α′
), and apply Algorithm 1.
5. (Local search) Starting from the grouping obtained in Step 4, systematically check all re-assignments
of units i ∈ 1, ..., N to groups g ∈ 1, ..., G (for g 6= gi), updating gi when the objective func-
tion decreases; stop when no further re-assignment improves the objective function.
Let the returning grouping be γ′′
.
6. If the objective function using γ′′
improves relative to the one using γ∗, then set γ∗ = γ′′
and go
to Step 2; otherwise, set n = n+ 1 and go to Step 7.
7. If n ≤ nmax, then go to Step 3; otherwise go to Step 8.
8. If time elapsed > timemax, then Stop; otherwise go to Step 2.
Algorithm 2 combines two different search technologies. First, a local search (Step 5) guarantees
that a local optimum is attained, in the sense that the solution cannot be improved by re-assigning
any single individual to a different group. Notice that solutions of Algorithm 1 do not necessarily
correspond to local minima in this sense. Secondly, re-assigning several randomly selected units into
randomly selected groups (Step 3) allows for further exploration of the objective function. This is done
by means of neighborhood jumps of increasing size, where the maximum size of the neighborhood nmax
12
is chosen by the researcher. Local search allows to get around local minima that are close to each
other, whereas random jumps aim at efficiently exploring the objective function while avoiding to get
trapped in a valley.
Lastly, unlike in Algorithm 1 the termination condition in Algorithm 2 depends on a time limit
timemax set by the researcher. The algorithm may also be run using different starting parameter values,
even though the choice of starting values tends to matter much less in this case. Algorithm 2 thus
depends on three parameters: the number of starting values (Ns), the maximum size of neighborhoods
(nmax), and the time limit (timemax).
3.2 Numerical performance
Tables 1 and 2 show the results of the computation of the GFE estimator on the cross-country panel
dataset that we use in the empirical application. The dataset is described in Section 6, but for now
it is enough to keep in mind its dimensions: N = 90, T = 7, and two covariates (including a lagged
outcome). We show the value of the objective as well as computation time for both algorithms, and
for G = 2, 3, and 10. In addition, we show the results for the first 30 countries, the first 60 countries
(alphabetically ordered), and all 90 countries in the dataset, respectively.
Table 1 suggests that the simple iterative algorithm performs well when the number of groups is
small. Algorithms 1 and 2 yield the same solution (that is, the same objective and optimal grouping)
in all configurations of the data. In contrast, Table 2 shows that Algorithm 2 improves on Algorithm
1 when the number of groups gets larger (G = 10).
When G = 10 and N = 30, running the iterative algorithm using 1000 starting values yields a
non-optimal solution. When all N = 90 countries are included in Table 2, even 1000, 000 different
starting values and a running time of approximately one hour is not enough to get to the optimal
solution. In contrast, Algorithm 2 is able to improve the objective after only four minutes of search
(7.749 versus 7.762, respectively). Interestingly, running the algorithm during 36 hours yields exactly
the same objective and grouping.
Despite these remarkable results, one concern is that even the best heuristic methods can lead to
non-optimal solutions. To assess whether the solutions of Algorithm 2 are optimal in Tables 1 and 2,
we make use of– and extend– exact solution algorithms for the minimum sum-of-squares partitioning
problem. New methods have recently been proposed to compute globally optimal solutions in this
challenging problem,14 including Brusco (2006)’s repetitive branch and bound algorithm, and Aloise
et al. (2009)’s column generation algorithm.
In the “exact” columns of Tables 1 and 2 (indicated with two or three stars) we report the objective
function obtained when applying one of these exact algorithms to the vector of residuals yit − x′itθ,
where θ is previously computed using our best heuristic (Algorithm 2). We see that the objective and
14It has been proved that problem (11) may be solved exactly in O(NGT+1) operations (Inaba et al., 1994).
13
Table 1: Numerical performance (G = 2, 3)
G = 2
Alg. 1 (1000) Alg. 2 (10;10;10) Exact
Value time Value time Value
N = 30 6.159 .6 6.159 2.1 6.159∗
N = 60 13.209 .9 13.209 7.6 13.209∗
N = 90 19.846 1.3 19.846 18.2 19.846∗
G = 3
Alg. 1 (1000) Alg. 2 (10;10;10) Exact
Value time Value time Value
N = 30 4.913 .6 4.913 6.1 4.913∗
N = 60 10.934 1.1 10.934 16.7 10.934∗∗
N = 90 16.598 1.7 16.598 38.4 16.598∗∗
Note: Balanced panel dataset from Acemoglu et al. (2008), T = 7, two covariates. Results for Algorithm 1 (Ns),
with Ns randomly chosen starting values; and for Algorithm 2 (Ns;nmax; timemax), with Ns starting values,
maximum size of neighborhoods nmax, and maximum time timemax. The value of the final objective and CPU
time (in seconds) are indicated. In the “exact” column, ∗∗ refers to Brusco (2006)’s exact branch and bound
algorithm for given θ, and ∗ refers to our extension of Brusco’s algorithm that allows for covariates.
Note: See note to Table 1. In the “exact” column, ∗∗∗ refers to Aloise et al. (2009)’s exact column generation
algorithm for given θ.
14
grouping coincide with the one identified by Algorithm 2 in all cases, including when G = 10. This
provides very encouraging evidence on the performance of our algorithm, which confirms previous
evidence obtained for minimum sum-of-squares partitioning (Brusco and Steinley, 2007).
In addition, we were able to extend Brusco (2006)’s repetitive branch and bound algorithm to
allow for covariates.15 Although our current implementation is limited to a small number of groups
(G = 2 for N ≤ 90, and G = 3 for N = 30) it yields the same solution as the one obtained using
the heuristics; see the results indicated with one star in Table 1. This formally demonstrates that our
heuristic algorithm has correctly identified the global minimum in these cases.
Overall, this section suggests that the computation problem for GFE is challenging, yet not im-
possible, thanks to recent advances in the clustering literature. Our main algorithm (Algorithm 2)
delivers fast and reliable solutions, and we have provided evidence that the solution is globally optimal
in datasets of moderate size. Assessing the performance of our algorithm in larger datasets is a natural
next step.
Finally, it is worth pointing out that research on exact computation algorithms is still in progress.
Recent research for solving problem (11) has shown that sophisticated interior point methods can
deliver exact solutions in competitive time in several large instances.16 We view these approaches as
a potentially useful complement to heuristic methods in order to compute the GFE estimator.
4 Asymptotic properties
In this section we characterize the asymptotic properties of the grouped fixed-effects estimator as N
and T tend to infinity simultaneously. We provide conditions under which estimated groups converge
to their population counterparts, and the bias of the GFE estimator shrinks to zero at a faster-than-
polynomial rate as T tends to infinity. This implies that the estimator is asymptotically equivalent
to an infeasible least-squares target, even when T diverges (polynomially) more slowly than N . As a
practical implication, the researcher will be able to conduct inference treating the estimated groups
as if they were the true ones.
15Brusco’s algorithm is available at: http://mailer.fsu.edu/∼mbrusco/bbwcss.for. The extension of the algorithm that
allows for covariates is available from the authors upon request.16While Brusco (2006)’s repetitive branch and bound algorithm computed the global minimum in (11) in Fisher’s Iris
data (N = 150, T = 4) for as much as G = 10 groups, du Merle et al. (2001) and more recently Aloise et al. (2009)
computed exact solutions in datasets of dimensions up to N = 2310 and T = 19, for G = 250 groups. Note that the
algorithm of Aloise et al. (2009) that we used in Table 2 delivered the global optimum in 1.7 seconds only.
In this first part we set the framework and provide some intuition for the main results. We consider
the following data generating process:
yit = x′itθ0 + α0
g0i t+ vit, (12)
where g0i ∈ 1, ..., G denotes group membership, and where the 0 superscripts refer to true parameter
values. We assume for now that the number of groups G = G0 is known, and we defer the discussion
on estimation of the number of groups until the next section.
Let(θ, α
)be the infeasible version of the GFE estimator where group membership gi, instead of
being estimated, is fixed to its population counterpart g0i :
(θ, α
)= argmin
(θ,α)∈Θ×AGT
N∑
i=1
T∑
t=1
(yit − x′itθ − αg0i t
)2. (13)
This estimator can be understood as the least-squares estimator in the pooled regression of yit on xit
and the interactions of population group dummies and time dummies.
The main result of this section establishes that, under suitable conditions, estimating the groups
does not affect the asymptotic properties of the grouped fixed-effects estimator. More precisely, we
show that the GFE estimator is asymptotically equivalent to the infeasible least-squares target(θ, α
)
as N and T tend to infinity and, for some ν > 0, N/T ν → 0. In particular, this allows T to grow
considerably more slowly than N (when ν ≫ 1). Before discussing the general case of model (12), we
start by providing an intuition in a simple case.
Intuition in a simple case. Let us consider a simplified version of model (12) in which group-
specific effects are time-invariant, θ0 = 0 is known (no covariates), vit are i.i.d. normal (0, σ2), and
G = G0 = 2. The model is thus:
yit = α0g0i
+ vit, g0i ∈ 1, 2, vit ∼ iidN (0, σ2). (14)
We further assume that α01 6= α0
2, and importantly that the distance between the two remains positive
as the sample size increases. Our asymptotic results do not hold uniformly with respect to the values
of the group-specific parameters, and require groups to be well-separated. In the next section we will
discuss a case where this condition fails. In addition we take α01 < α0
2 without loss of generality.
In finite samples, there is a non-zero probability that estimated and population group membership
do not coincide. It follows from (3) that the probability of misclassifying an individual who belongs
to group 1 into group 2 is:
Pr(gi(α0)= 2∣∣ g0i = 1
)= Pr
(T∑
t=1
(α01 + vit − α0
2
)2<
T∑
t=1
(α01 + vit − α0
1
)2)
= Pr
(vi >
α02 − α0
1
2
).
16
Provided α is consistent for α0, the misclassification probability can thus be approximated by:
Pr(gi (α) = 2
∣∣ g0i = 1)≈ 1− Φ
(√T
(α02 − α0
1
2σ
)). (15)
The GFE estimator α suffers from an incidental parameter bias due to the fact that the number of
g0i parameters tends to infinity with N . For fixed T , g0i is not consistently estimated, and as a result
α is inconsistent as N tends to infinity.17 Nevertheless, (15) implies that the group misclassification
probability tends to zero at an exponential rate, which intuitively means that the incidental parameter
problem vanishes very rapidly as T increases.
In this simple model, it can easily be shown that, for g = 1, 2, the difference between αg and the
infeasible sample mean
αg =
∑Ni=1 1
g0i = g
yi∑N
i=1 1g0i = g
is exponential in T . Suppose now that, for some ν > 0, N/T ν → 0. It then follows that√NT (αg − αg)
tends to zero asymptotically, and hence that α and α have the same asymptotic distribution. Note
that this result is specific to models with discrete heterogeneity: when αi can take continuous values,
in contrast, biases due to the incidental parameter problem are typically of the O(1/T ) order, and
asymptotic equivalence with an unbiased infeasible target only holds if N/T → 0 (e.g., Nickel, 1981,
Hahn and Newey, 2004).
Extending the analysis of model (14) to a more general setup raises two main challenges. Consis-
tency is not straightforward to establish since, as N and T tend to infinity, both the number of group
membership variables gi and the number of group-specific time effects αgt tend to infinity, causing an
incidental parameter problem in both dimensions.18 Secondly, the argument leading to the exponential
rate of convergence of the misclassification probability (15) relies on the assumption that errors are
i.i.d. normal. In order to bound tail probabilities under more general conditions (e.g., non-normality),
approximations based on a central limit theorem are not sufficient. The analysis that we present next
addresses both challenges.
4.2 Main results
We start by showing consistency under the following assumptions.
Assumption 1 Let M > 0 be some constant.
a. Θ and A are compact subsets of RK and R, respectively.
17The properties of GFE for fixed T follow from a direct extension of Pollard (1981, 1982). The estimator converges
at a root-N rate to its probability limit. However, the latter does not coincide with the true parameter value in general
(Bryant and Williamson, 1978).18Note that the class of models considered in a recent paper by Hahn and Moon (2010) only covers time-invariant
discrete unobserved heterogeneity. So their results do not apply here.
17
b. E
(‖xit‖2
)≤M , where ‖·‖ denotes the Euclidean norm.
c. E (vit) = 0, and E(v4it)≤M .
d. For all g ∈ 1, ..., G:∣∣∣ 1NT
∑Ni=1
∑Tt=1
∑Ts=1 E
(vitvisα
0gtα
0gs
)∣∣∣ ≤M .
e.∣∣∣ 1NT
∑Ni=1
∑Tt=1
∑Ts=1 E (vitvisx
′itxis)
∣∣∣ ≤M .
f. 1N
∑Ni=1
∑Nj=1
∣∣∣ 1T∑T
t=1 E (vitvjt)∣∣∣ ≤M .
g.∣∣∣ 1N2T
∑Ni=1
∑Nj=1
∑Tt=1
∑Ts=1Cov (vitvjt, visvjs)
∣∣∣ ≤M .
h. Let xg∧g,t denote the mean of xit in the intersection of groups g0i = g, and gi = g. Let ρ be the
minimum eigenvalue of the following matrix, where the infimum is taken over all possible groupings
γ = g1, ..., gN:
infγ∈ΓG
1
NT
N∑
i=1
T∑
t=1
(xit − xg0i ∧gi,t
)(xit − xg0i ∧gi,t
)′.
Then plimN,T→∞
ρ = ρ > 0.
In Assumption 1.a we require the parameter spaces to be compact. It is possible to relax this
assumption and alternatively assume that the group-specific time effects α0gt have finite (fourth-order)
moments, as in Bai (2009). However, allowing the group effects to follow non-stationary processes
would require a different analysis, which is not considered in this paper. Similarly, we rule out non-
stationary covariates and errors in Assumptions 1.b and 1.c, respectively.
Weak dependence conditions are required in Assumptions 1.d to 1.g. These are conceptually similar
to assumptions commonly made in the literature on large factor models (Stock and Watson, 2002, Bai
and Ng, 2002). Note that Assumptions 1.d and 1.e allow α0gt and xit to be weakly exogenous. In
particular, this allows for lagged outcomes and general predetermined regressors. Assumptions 1.d,
1.e and 1.g impose conditions on the time-series dependence of errors (as well as covariates and time
effects), while Assumption 1.f restricts the amount of cross-sectional dependence. Note that these
assumptions are satisfied in the special case where vit are i.i.d. across units and time periods, and
where E(α0gtvit
)= 0 and E (xitvit) = 0.
Note also that Assumption 1.e may still be satisfied when errors and covariates are correlated to
each other. An important example for applied work is model (5), when estimated in deviations to
unit-specific means so as to remove time-invariant unit fixed-effects. In this case the assumption allows
for predetermined regressors; for example, it is satisfied if xit = yi,t−1 is a lagged outcome (see, e.g.,
Alvarez and Arellano, 2003). When covariates are endogenous and Assumption 1.e does not hold,
however, GFE leads to inconsistent estimates in general.
18
Lastly, Assumption 1.h is analogous to a full rank condition in standard regression models. We
require that xit shows sufficient variation over time and across individuals.19 As a special case, the
condition will be satisfied if xit is discrete and, for all g, the conditional distribution of (xi1, ., , , xiT )
given g0i = g has strictly more than G points of support. For example, if xit follows a non-degenerate
Bernouilli distribution, i.i.d in both dimensions, then (xi1, ..., xiT ) has 2T points of support, which
may well be larger than G + 1. Note also that Assumption 1.h allows for time-invariant regressors,
provided that their support is rich enough.
We have the following result, where for conciseness we denote as gi = gi
(θ, α
)the GFE estimates
of g0i , for all i.
Theorem 1 (consistency) Let Assumption 1 hold. Then, as N and T tend to infinity: θp→ θ0, and
1NT
∑Ni=1
∑Tt=1
(αgit − α0
g0i t
)2 p→ 0.
Proof. See Appendix A.
The consistency proof is complicated by the fact that the dimension of α diverges as T tends to
infinity. This prevents the adoption of standard techniques (e.g., Newey and McFadden, 1994) to prove
the result. Instead, we build on an insight from Bai (1994, 2009) and consider an auxiliary objective
function whose minimum is attained at(θ0, α0
). The strategy of the proof consists then in showing
that the difference between the GFE objective function and the auxiliary one becomes uniformly small
as N and T tend to infinity.
We now characterize the asymptotic distribution of the GFE estimator under the following as-
sumptions.
Assumption 2 Let a, b, c, d1, d2 > 0 be constants.
a. For all g ∈ 1, ..., G: plimN→∞1N
∑Ni=1 1g0i = g = πg > 0.
b. For all (g, g) ∈ 1, ..., G2 such that g 6= g: 1T
∑Tt=1
(α0gt − α0
gt
)2≥ c.
c. For all i ∈ 1, ..., N and all g ∈ 1, ..., G, vitt and vitα0gtt are zero-mean stationary and
strongly mixing processes with mixing coefficients that satisfy α[t] ≤ e−atd1 for all t.
d. Pr (|vit| > m) ≤ e1−(mb )
d2
for all i, t, and m > 0.
e. One of the two following conditions holds:
(i) xit has bounded support in RK .
(ii) ‖xit‖t satisfies the mixing and tail conditions of 2.c and 2.d above.
19Assumption 1.h is interestingly related to Assumption A in Bai (2009).
19
In contrast with consistency, we restrict the analysis of the asymptotic distribution to the case
where the G population groups are well-separated (Assumptions 2.a and 2.b). In general the properties
of the GFE estimator are different if group separation fails, for example when the number of groups in
the population is strictly smaller than the number of groups postulated by the researcher (i.e., when
G0 < G). In the next section we will come back to this important issue.
In Assumptions 2.c and 2.d we restrict the dependence and tail properties of vit, respectively.
Specifically, we assume that vit is α-mixing with a faster-than-polynomial decay rate, with tails also
decaying at a faster-than-polynomial rate. The process vitα0gt is assumed to have zero mean and to be
strongly mixing as well. Note that this strengthens the assumptions made in Assumption 1 regarding
time-series dependence. These conditions allow us to rely on exponential inequalities for dependent
processes (e.g., Rio, 2000) in order to bound misclassification probabilities.20 Finally, in Assumption
2.e we impose either of two conditions on covariates xit. In Part (i) we require that covariates have
bounded support. Alternatively, in Part (ii) we require that covariates satisfy dependence and tail
conditions similarly as vit. In particular, lagged outcomes (xit = yi,t−1) are covered under these
conditions.
The next result shows that the GFE estimator and the infeasible least squares estimator with known
population groups are asymptotically equivalent. Note that, because of invariance to re-labelling of
the groups, the results for group membership and group-specific effects are understood to hold given
a suitable choice of the labels (see the proof for details).
Theorem 2 (asymptotic equivalence) Let Assumptions 1.a-1.h, and 2.a-2.e hold. Then, for all δ > 0
and as N and T tend to infinity:
Pr
(sup
i∈1,...,N
∣∣gi − g0i∣∣ > 0
)= o(1) + o
(NT−δ
), (16)
and:
θ = θ + op
(T−δ
), and (17)
αgt = αgt + op
(T−δ
)for all g, t. (18)
Proof. See Appendix A.
It follows from Theorem 2 that the asymptotic distribution of the grouped fixed-effects estimator
and that of the infeasible least squares estimator coincide if, for some ν > 0, N/T ν tends to zero as
20It is possible to relax Assumptions 2.c-2.e and assume that vit, vitα0gt, and possibly ‖xit‖, are strongly mixing
with a polynomial decay rate, and that their marginal distributions have polynomial tails, i.e. that α[t] ≤ at−d1 , and
Pr (|vit| > m) ≤ m−d2 for some constants a ≥ 1, d1 > 1, and d2 > 2. Under these weaker assumptions, it may be shown
that the GFE estimator is unbiased to order q, provided that (d1+1)d2d1+d2
≥ 4q + 1.
20
N and T tend to infinity simultaneously. For example we have, using (17):
√NT
(θ − θ
)= op
(N
12T
12−δ),
which is op(1) as soon as δ ≥ (ν + 1)/2. In addition, under these relative rates of N and T the
estimated groups are uniformly consistent for the population ones, in the sense that:
supi∈1,...,N
∣∣gi − g0i∣∣ p→ 0. (19)
Note that these relative rates allow T to increase (polynomially) more slowly than N . In contrast,
the large N,T asymptotic analysis of FE estimators is typically done assuming that N/T → Cst, or
that N/T 3 → 0 in the case of bias-reduced estimators (e.g., Arellano and Hahn, 2006). Asymptotic
equivalence as N/T ν → 0 is the consequence of the fact that, unlike most fixed-effects or interactive
fixed-effects estimators, the GFE estimator is unbiased to any (polynomial) order of magnitude relative
to the infeasible least-squares target.
The following assumptions allow to simply characterize the asymptotic distribution of the least-
squares estimator(θ, α
), where we denote as xgt the mean of xit in group g0i = g.
Assumption 3
a. For all i, j and t: E (xjtvit) = 0.
b. There exist positive definite matrices Σθ and Ωθ such that:
Σθ = plimN,T→∞
1
NT
N∑
i=1
T∑
t=1
(xit − xg0i t
)(xit − xg0i t
)′
Ωθ = plimN,T→∞
1
NT
N∑
i=1
N∑
j=1
T∑
t=1
T∑
s=1
E
[vitvjs
(xit − xg0i t
)(xjs − xg0j s
)′].
c. As N and T tend to infinity:
1√NT
N∑
i=1
T∑
t=1
(xit − xg0i t
)vit
d→ N (0,Ωθ) .
d. For all (g, t):
plimN→∞
1
N
N∑
i=1
N∑
j=1
E(1g0i = g0j = gvitvjt
)= ωgt > 0.
e. For all (g, t), and as N and T tend to infinity:
1√N
N∑
i=1
1g0i = gvit d→ N (0, ωgt) .
21
Assumptions 3.a-3.c imply that the least-squares estimate θ has a standard asymptotic distribution.
In particular, Assumption 3.a ensures that the estimator has no asymptotic bias. Note that this
condition is satisfied if xit is strictly exogenous or predetermined and observations are independent
across units. As a special case, lagged outcomes may thus be included in xit. Note however that this
assumption rules out lagged outcomes in model (5) with additive fixed-effects. Indeed, in deviations
to unit-specific means we have: E[(vit − vi)
(yi,t−1 − yi,−1
)]6= 0, and the least-squares estimator
suffers from an O(1/T ) bias.21 Lastly, Assumptions 3.d-3.e similarly ensure that αgt has a standard
asymptotic distribution.
We have the following result.
Corollary 1 (asymptotic distribution) Let Assumptions 1.a-1.h, 2.a-2.e, and 3.a-3.e hold, and let N
and T tend to infinity such that, for some ν > 0, N/T ν → 0. Then we have:
√NT
(θ − θ0
)d→ N
(0,Σ−1
θ ΩθΣ−1θ
), (20)
and, for all (g, t):√N(αgt − α0
gt
) d→ N(0,ωgt
π2g
), (21)
where πg is defined in Assumption 2, and where Σθ, Ωθ, and ωgt are defined in Assumption 3.
Proof. See Appendix A.
We end this section with two remarks. Asymptotically accurate group classification, as established
in Theorem 2, has practical consequences. As an example, suppose one wants to fit a parametric
model (e.g., a multinomial logit model), indexed by a parameter vector ξ, to the estimated group
probabilities:
ξ = argmaxξ
N∑
i=1
G∑
g=1
1 gi = g ln (pg (xi; ξ)) ,
where pg(x; ξ) are the parametrically specified group probabilities. Then, under similar conditions
as in Theorem 2, ξ will be asymptotically equivalent to the following infeasible maximum likelihood
estimator:
ξ = argmaxξ
N∑
i=1
G∑
g=1
1g0i = g ln (pg (xi; ξ)) .
This implies that parameter estimates (and their standard errors) that treat the estimated groups as
data will be asymptotically valid.
21Note that the GFE estimates of group membership are consistent in model (5) if the conditions of Theorem 2 are
satisfied. In the presence of lagged outcomes in (5), one could thus estimate g1, ..., gN using GFE, and in a second step
estimate the other parameters using any dynamic panel data estimator conditional on the estimated group dummies and
time dummies. The large N,T properties of this type of two-step estimators would require a separate analysis.
22
Lastly, note that the equivalence result in Theorem 2 still holds when considering a penalized
version of the grouped fixed-effects estimator that incorporates non-dogmatic prior information, as we
show in Appendix B. In FE models, adding prior information on the individual effects has generally
a first-order effect on the bias of the estimator (Arellano and Bonhomme, 2009). In contrast, in
models where unobserved heterogeneity is discrete, and under the conditions of Theorem 2, adding
non-dogmatic prior information has no effect on the asymptotic distribution of the estimator.
5 Practical issues
In this section we discuss two important practical issues: estimation of the covariance matrices and
estimation of the number of groups. In addition, we show the results of a small simulation experiment
aimed at assessing the finite sample performance of our estimator, as well as that of our inference
methods and choice of the number of groups.
5.1 Estimating covariance matrices
Here we discuss estimation methods for the covariance matrices appearing in Corollary 1 under different
assumptions.
Group-specific time effects. If observations are assumed independent across individual units the
variance of αgt for all g, t can be estimated using the White formula:
Var (αgt) =
∑Ni=1 1 gi = g v2it(∑Ni=1 1 gi = g
)2 , (22)
where vit = yit − x′itθ − αgit are the estimated GFE residuals.
Common parameters. Following Corollary 1, we estimate the asymptotic variance of θ as follows:
Var(θ)=
Σ−1θ ΩθΣ
−1θ
NT, (23)
where, denoting as xgt the mean of xit in group gi = g, we take:
Σθ =1
NT
N∑
i=1
T∑
t=1
(xit − xgi,t
) (xit − xgi,t
)′,
and where Ωθ is a consistent estimate of the matrix Ωθ.
In the presence of serial correlation, but in the absence of correlation across units, one may use the
truncated kernel method of Newey and West (1987) in order to construct an estimator Ωθ, as in Bai
23
(2003). Alternatively, one may use the following formula clustered at the individual level (Arellano,
1987):
Ωθ =1
NT
N∑
i=1
T∑
t=1
T∑
s=1
vitvis(xit − xgi,t
) (xis − xgi,s
)′.
The properties of Arellano (1987)’s formula in FE models as N and T tends to infinity are studied
in Hansen (2007). Below we show numerical evidence on the finite sample performance of the estimator
(23) of the variance of the GFE estimator.
Lastly, note that the assumptions of Corollary 1 allow for weak dependence in the cross-sectional
dimension. However, the clustered variance formula is generally invalid in that case. A robust alterna-
tive is to follow Bai and Ng (2006), and to construct a partial sample estimator Ωθ based on a random
sample of size n << min(N,T ).
5.2 Unknown number of groups
The asymptotic results of Section 4 were derived under the assumption that the true number of groups
G0 is known. In practice, however, this is rarely the case. Here we relax this assumption and let G be
the (possibly incorrect) number of groups postulated by the researcher.
Incorrect number of groups: a simple case. Misspecification of the number of groups has
different effects on common parameter estimates, depending on whether the postulated number of
groups is above or below the true one.
When G < G0, the GFE estimator θ is generally inconsistent for θ0 if the unobserved effects are
correlated with the observed covariates. The inconsistency arises because of omitted variable bias.
In contrast, when G > G0 common parameters θ remain consistent for θ0 under the conditions of
Theorem 1, since the proof of the theorem is unaffected in this case. However, the group-specific
effects suffer from a substantial small-T bias, as the following simple example illustrates.
Proposition 1 Let us consider the model:
yit = x′itθ0 + α0
g0i+ vit, vit ∼ iidN (0, σ2), (24)
where the true number of groups is G0 = 1, and where α0 = α01 denotes the true value of α.
Let(θ, α
)be the GFE estimator of (θ0, α0) with G = 2 groups. Then, as T is kept fixed and N
tends to infinity we have: θp→ θ0, and αg
p→ α0 ± σ√
2πT
, for g = 1, 2.
Proof. See Appendix A.
In this example, the data generating process is homogeneous (G0 = 1), but the researcher estimates
two groups (G = 2). The proof of Proposition 1 shows that, asymptotically, the two estimated groups
24
are solely based on random errors (depending on whether vi ≥ 0). Given that the spurious groups
are independent of covariates, their presence does not bias the GFE estimator of θ0. In fact, allowing
for a larger number of groups than the true one in GFE estimation may be thought of as including
(G−G0) irrelevant regressors– uncorrelated with the covariates of interest– in a linear regression.
A similar intuition applies to interactive fixed-effects models: Moon and Weidner (2010b) show
that the asymptotic distribution of the interactive FE estimator with G ≥ G0 factors is identical to
that of the estimator based on the correct number of factors. We conjecture that a similar result
applies to the GFE estimator in model (1). However, a formal proof of this conjecture is beyond the
scope of this paper.
In contrast with common parameters, although the group effects α1 and α2 are both consistent to
α0 as T tends to infinity, they suffer from a bias term of order Op(1/√T ) for small T , which is one
order of magnitude larger than the usual Op(1/T ) order in FE panel data models. The σ√
2πT
term
in Proposition 1 is simply the mean of a truncated normal (0, σ2/T ) (i.e., the mean of vi truncated at
zero).
Estimating the number of groups. To consistently estimate the number of groups G0 we rely on
the connection with the analysis of large factor models and interactive fixed-effects panel data models.
We consider the following class of information criteria:
I (G) =1
NT
N∑
i=1
T∑
t=1
(yit − x′itθ
(G) − α(G)it
)2+GhNT , (25)
where(θ(G), α(G)
)is the grouped fixed-effects estimator with G groups. The estimated number of
groups is then:
G = argminG∈1,...,Gmax
I (G) , (26)
where Gmax is an upper bound on G0, which is assumed known in order to derive the asymptotic
properties.
Following the arguments in Bai and Ng (2002) and Bai (2009), it can be shown that the estimated
number of groups G is consistent for G0 if, as N and T tend to infinity, hNT tends to zero and
min(N,T )hNT tends to infinity. The first condition ensures that G ≥ G0 with probability approaching
one, while the second condition guarantees that G ≤ G0.
As an example, let us consider the following Bayesian Information Criterion (BIC):22
BIC (G) =1
NT
N∑
i=1
T∑
t=1
(yit − x′itθ
(G) − α(G)it
)2+ σ2
GT +N +K
NTln(NT ), (27)
22Given that unobserved heterogeneity is discrete, there is some ambiguity on how to define the number of parameters
in the grouped fixed-effects approach. In (27) we have simply added the number of group-specific time effects (that
is, GT ), the number of common parameters (K), and the number of group membership variables gi (that is, N). In
Appendix C we report simulation results using (27), as well as using an alternative choice with a stronger penalty.
25
where σ2 is a low bias estimate of the variance of vit.23 One easily sees that the BIC estimate G
provides an upper bound on G0 asymptotically if (lnT )/N → 0. In addition, G is consistent for G0 if
N and T tend to infinity at the same rate. In contrast, if T tends to infinity more slowly than N so
that T/N tends to zero, the BIC criterion (27) provides a conservative, possibly inconsistent, estimate
of G0.
5.3 A small-sample exercise
In the last part of this section, we study the suitability of our main asymptotic results as a guide for
small sample inference. We do this by means of a Monte Carlo exercise on simulated data, which we
design to approximate the cross-country dataset that we will use in the empirical application.
Specifically, we consider the same sample size: N = 90 units and T = 7 periods. For a given number
of groups, the data generating process follows model (12) where xit = (yi,t−1, xit) contains a lagged
outcome and a strictly exogenous regressor, and where the process xit is taken from the log-income per
capita data. For this specification, we first estimate the model on the empirical dataset using grouped
fixed-effects. Then, we fix the parameters of the DGP: θ0, α0 and all the group membership variables
g0i , to their estimated GFE values. Lastly, the error terms are generated as i.i.d. normal draws across
units and periods with variance equal to the mean of squared GFE residuals.
We start by showing the mean of the GFE estimator across 1, 000 Monte-Carlo simulations: Table
3 shows that the bias on the autoregressive parameter ranges between -4% (for G = 3, .391 when the
truth is .407) and 33% (for G = 5, .339 versus .255), while the bias on the second coefficient is always
smaller than 10%. The table also shows the “long-run” coefficient of xit (divided by one minus the
autoregressive parameter), whose bias ranges between 6% and 18%.
The last column in Table 3 shows the average misclassification frequency across simulations.24
When G = 3 or 5, units are well classified in approximately 90% of cases. When G = 10, however, the
frequency of correct classification drops to 65%. Nonetheless, the bias of the GFE estimator remains
rather small. This suggests that our asymptotic theory for θ may provide a reasonable guide for finite
sample performance, even in situations in which G is not small relative to the sample size. In the
conclusion, we shall comment on the possibility to modify the asymptotic analysis in order to allow G
to increase together with N and T at a suitable rate.
We next turn to inference. Table 4 reports the standard deviation of the GFE estimator of θ across
23A possibility is to estimate θ and α using grouped fixed-effects with Gmax groups, and to compute:
σ2 =
1
NT −GmaxT −N −K
N∑
i=1
T∑
t=1
(yit − x
′
itθ − αgi(θ,α)t
)2.
24The misclassification frequency is computed as 1N
∑N
i=1 1gi 6= g0i . To deal with invariance to relabelling, we take
the minimum of this frequency across all G! permutations of group indices. When G = 10 this computation results
prohibitive, so we take the minimum over 100, 000 randomly generated permutations.
26
Table 3: Bias of the GFE estimator
θ1 (coeff. yi,t−1) θ2 (coeff. xit)θ2
1−θ1Misclassified
True GFE True GFE True GFE
G = 3 .407 .391 .089 .099 .151 .163 9.65%
G = 5 .255 .339 .079 .083 .107 .126 9.20%
G = 10 .277 .286 .075 .078 .104 .110 34.84%
Note: The columns labelled “GFE” refer to the mean of GFE parameter estimates across 1, 000 simulations.
Algorithm 2– with parameters (5; 10; 5)– was used for computation. The last column shows the average of the
misclassification frequency (gi 6= g0i ) across simulations. Errors are i.i.d. standard normal.
Table 4: Standard deviation of the GFE estimator
θ1 (coeff. yi,t−1) θ2 (coeff. xit)θ2
1−θ1
Asymptotic Monte Carlo Asymptotic Monte Carlo Asymptotic Monte Carlo
G = 3 .035 .043 .0094 .0137 .013 .021
G = 5 .044 .058 .0098 .0112 .014 .022
G = 10 .037 .059 .0075 .0103 .007 .015
Note: See the notes to Table 3. The columns labelled “Asymptotic” and “Monte Carlo” refer to the estimates
based on the clustered variance formula (23), and to the standard deviation across 1, 000 simulations, respectively.
Monte Carlo simulations, and average values of the clustered formula (23). Although the order of
magnitude in the two expressions is similar, the results show that the clustered formula systematically
underpredicts the variability of the GFE estimator. This suggests that group misclassification may
have a sizable effect on inference in small samples. Studying the properties of resampling methods
such as the residual bootstrap is an interesting possibility.
Finally, in Appendix C we show the results of several additional exercises. We estimate a natural
alternative, the interactive fixed-effects estimator of Bai (2009) with 3 factors, when the DGP has
G = 3 groups. Although the interactive FE estimator is consistent as N and T tend to infinity, our
results show that it suffers from a very substantial finite sample bias, much larger than the bias of
the GFE estimator on this (relatively small) sample. Then, given that the asymptotic behavior of
the GFE estimator crucially depends on tail and dependence properties of errors, we estimate a non-
normal specification with dependent errors, finding similar results as in the main exercise. We also
report results for the group-specific time effects. Lastly, we provide evidence on the accuracy of the
BIC criterion (27) to estimate the number of groups on the simulated data.
27
6 Application: income and (waves of) democracy
In this last part of the paper we use the grouped fixed-effects approach to study the relationship
between income and democracy across countries.
6.1 The empirical setup
The statistical association between income and democracy is an important stylized fact in political
science and economics (Lipset 1959, Barro 1999). In an influential paper, Acemoglu, Johnson, Robin-
son and Yared (2008) emphasize the importance to account for factors that simultaneously affect
both economic and political development. Using country-level panel data, they document that the
positive effect of income on democracy disappears when including country-specific fixed-effects in the
regression.
Acemoglu et al. (2008) argue that these results are consistent with countries having embarked on
divergent paths of economic and political development at certain points in time. Examples of such
events, or “critical junctures”, could be the end of feudalism, the industrialization age, or the process
of colonization (as in Acemoglu, Johnson and Robinson, 2001). In this perspective, the inclusion of
fixed-effects in the regression is meant to capture these highly persistent long-run historical effects.
Table C4 in Appendix C replicates the main specification from Acemoglu et al.: a fixed-effects
regression of democracy on lagged income per capita, using lagged democracy and time dummies
as controls.25 Democracy is measured according to the Freedom House indicator, and log-GDP per
capita is taken from the Penn World tables. All data are taken at the five-year frequency. Both the
balanced sample, which covers 90 countries on the period 1970 − 2000, and the unbalanced panel,
which covers 150 countries on the period 1960 − 2000, show similar results. According to the pooled
OLS regressions, there is a statistically significant association between income and democracy. The
point estimates imply that a 10% increase in income per capita is associated with an increase in the
Freedom House score of 2.5%.26 However, in both datasets, the FE estimates are small or negative,
and insignificant from zero.
There are reasons to think that FE may not be the most appropriate methodology in order to draw
conclusions regarding the observed and unobserved determinants of democracy. On the one hand, a
large literature in political science emphasizes time-varying determinants of political regimes (e.g.,
Przeworski et al., 2000). At odds with this evidence, the fixed effects are assumed not to vary within
the sample period. On the other hand, the FE estimation results– common parameter estimates, as
well as country-specific fixed-effects– are imprecise given the small within-country variance of income
(6% of the total variance of income in the balanced sample), and the short length of the panel (7
25All data in this section are taken from the files of Acemoglu et al. (2008): http://economics.mit.edu/files/500026To assess the magnitude of this effect, note that the Freedom House measure is normalized to lie between zero and
one, and that its mean and standard deviation in the balanced sample are .55 and .37, respectively.
with an additive specification for the unobserved country-specific determinants of democracy: αit =
ηi+ δt. We shall compare and contrast the empirical results using several alternative specifications for
αit. In addition to the estimates of θ1 and θ2 and the implied cumulative income effect θ2/(1 − θ1),
we are also interested in estimating and interpreting the country unobservables αit.
Our main estimation results correspond to the baseline grouped fixed-effects specification: αit =
αgit, in which we allow for unrestricted group-specific time patterns of heterogeneity for several values
of the number of groups G. In addition, we consider two other specifications: one that combines group-
specific time-varying heterogeneity with country-specific time-invariant effects, that is αit = αgit + ηi;
and another one that contains two different layers of grouped heterogeneity: αit = αgi1t+ηgi1,gi2 . These
alternative specifications will provide us with novel insights regarding the unobserved determinants of
democracy and their evolution over time.
Allowing for time-varying group-specific patterns of heterogeneity in this context is empirically
motivated by the strong evidence of clustering of regime types and transitions, across time and space,
documented in the political science literature (e.g., Gleditsch and Ward, 2006, Ahlquist and Wibbels,
2012). Moreover, a conceptual motivation is given by Samuel Huntington’s influential work on the
“third wave” of democratization.
Huntington (1991) emphasizes the importance of international and regional factors as drivers of
transitions to democracy and autocracy, resulting in groups of countries making transitions at similar
points in time; that is, in “waves” of democratization.27 Along with other examples, he mentions the
influence of the US administration in the 1970s and changes in the Soviet Union in the early 1980s,
the influence of the European Union in the late 1970s, or changes in the Catholic Church following
the second Vatican council, as possible drivers of the clustering of transitions towards democracy that
occurred between 1974 and 1990.
Huntington’s arguments are consistent with the grouped fixed-effects model: for example gi could
denote being predominantly Catholic, and αgt could be the effect of the influence of the Catholic
Church on the political evolution of the country. However, our estimation framework is agnostic about
the causes of the “waves” of democracy, as it uncovers heterogeneous patterns of political evolution
from the data. In particular, our framework provides a natural starting point to assess how well the
27Huntington (1991) distinguishes three waves of democratization: the first one starting in the 1820s in the US and
ending with World War I, the second wave lasting between the end of World War II and the early 1960s, and the third
wave starting with the Portuguese revolution in 1974. The first two waves were followed by two “counterwaves”, in the
1930s and the 1960s, respectively. According to this typology it is still unclear whether the recent Arab spring will be
the start of a “fourth wave” of democratization (Diamond, 2011).
29
political and economic evolution of countries over time fits different theories of democratization.
6.2 Results
Here we discuss the estimates of the coefficients of income and lagged democracy, and of the grouped
patterns of heterogeneity and group membership. Then we assess the robustness of our results when
using alternative specifications and data. Lastly, we run ex-post regressions of the estimated groups
in order to explore why the estimated paths differ across countries.
6.2.1 Income effect
We start by presenting the estimates of the coefficients of income and lagged democracy for the baseline
grouped fixed-effects model (1). We report the results for the balanced subsample. Results for the
unbalanced sample are qualitatively similar, and are summarized in Section 6.2.3 below.
Figure 1 plots the point-estimates and standard errors of income and democracy coefficients for
different values of the number of groups G.28 The right panel shows that the implied cumulative
income effect θ2/(1 − θ1) sharply decreases from .25 in OLS to .10 for G = 5, and remains almost
constant as G increases further. The left and middle panels show that this pattern is mostly driven
by a decrease in the coefficient of lagged democracy. This is consistent with unobserved country
heterogeneity being positively correlated with (lagged) democracy, causing an upward bias in OLS.29
According to our estimates, the cumulative income effect is statistically significant. The Monte
Carlo experiments reported in Section 5 suggest that the clustered formula that we use to compute
standard errors may understate the finite sample variability. However, this underestimation is unlikely
to affect the significance of the results. For example, for G = 10, Table C2 in Appendix C shows that
residual bootstrap-based standard errors are .013, versus .009 for the clustered standard errors value.
This suggests that, unlike FE, the GFE estimates remain strongly significant even for rather large
values of G.30
Though statistically significant, the point estimates of the cumulative income effect in Figure 1
are quantitatively small: only 40% of the pooled OLS estimate when G ≥ 5. Moreover, we will see in
28All estimates were computed using Algorithm 2– with parameters (10; 10; 10). We performed extensive checks of
numerical accuracy, some of which are described in Section 3.29The implied cumulative effect of income shown in Figure 1 is almost identical to the estimated income effect when
using a specification that only controls for lagged GDP per capita (and does not include lagged democracy). Results are
available upon request.30The reason for this is that the within-group (that is, within-(gi, t)) variance of income remains sizable: it is 65% of
the total income variance when G = 3, 48% when G = 10, and still 43% when G = 15. This is substantially larger than
the within-country variance of income (6%). In contrast, the within-group variance of democracy is 10% when G = 15,
whereas the within-country variance is 26%. This difference arises because the groups are estimated so that they fit the
outcome (democracy), but not necessarily the regressor (income).
30
Figure 1: Income and democracy, GFE
Lagged democracy Lagged income Cumulative income effect
(θ1) (θ2) ( θ21−θ1
)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 150
0.2
0.4
0.6
0.8
1
number of groups G
lag
ge
d d
em
ocra
cy c
oe
ffic
ien
t
1 2 3 4 5 6 7 8 9 1011121314150
0.025
0.05
0.075
0.1
0.125
0.15
number of groups G
lag
ge
d in
co
me
co
eff
icie
nt
1 2 3 4 5 6 7 8 9 10 11 12 13 14 150
0.1
0.2
0.3
0.4
number of groups G
imp
lie
d c
um
ula
tive
in
co
me
eff
ect
Note: Balanced panel from Acemoglu et al. (2008). The x-axis shows the number of groups G used
in estimation, the y-axis reports parameter values. 95%-confidence intervals clustered at the country
level are shown in dashed lines.
Section 6.2.3 that the association between income and democracy disappears in a specification that
combines both time-varying grouped effects and time-invariant country-specific effects.
Lastly, the values reported in Table C5 in Appendix C show that the objective function decreases
steadily as G increases: by almost 50% when G = 5 compared to OLS, and by 75% when G =
13. Interestingly, comparison with the last row of the table shows that the objective function of
grouped fixed-effects is lower than the one of fixed-effects as soon as G ≥ 3. This suggests that a
substantial amount of cross-country heterogeneity is time-varying.31 We now document these time-
varying patterns.
6.2.2 Grouped patterns
The estimates of the unobserved determinants of democracy reveal heterogeneous, time-varying pat-
terns. Figure 2 shows the estimates of the groups-specific time effects and reports group membership
by country on a World map, for G = 4. The bottom panel shows the parameter estimates αgt, as well
as average democracy and (lagged) income per group over time.
31Another result of Table C5 is that G = 10 is optimal according to BIC. Recall from Section 5 that this criterion
provides a conservative estimate of the number of groups if T grows at a slower rate than N . Note also that the GFE
estimates in Figure 1 do not vary much between G = 5 and G = 15. According to the discussion in Section 5.2, this is
consistent with the true number of groups being actually smaller than 10. Optimal choice of G in practice is a notoriously
difficult problem in related contexts (e.g., mixture and factor models), which deserves further study.
31
The figure shows that two of the four groups experience stable paths of democracy over time,
albeit at very different levels. The first group– which corresponds to high-income, high-democracy
countries– includes the US and Canada, most of Continental Europe, Japan and Australia, but also
India and Costa Rica. The second group–low income, low democracy– includes a large share of North
and Central Africa, China, and Iran, among others. Note that these two groups, which together
account for 59 of the 90 countries, are broadly consistent with an additive fixed-effects representation,
as their grouped effects α1t and α2t are almost constant over time. In addition, group membership
is strongly correlated with log-GDP per capita, consistently with the presence of an upward omitted
variable bias in the cross-sectional regression of democracy on income.
While the first two groups of countries are consistent with FE, the other two are not. The third
group experiences a clear transition to democracy in the first part of the sample period: the mean
Freedom House score increases from .20 in 1970 to almost .90 in 1990. This group includes a large share
of Latin America, Greece, Spain and Portugal, Thailand and South Korea, in total 13 countries, with
an intermediate level of GDP per capita. The fourth group makes a later transition to democracy: its
average Freedom House score increases from .20 to .75 between 1985 and 2000. This group includes 18
countries, among which are a large part of West and South Africa, Chile, Romania, and Philippines.
These are low-income countries, whose GDP per capita is similar on average to that of group 2.
The grouped patterns in Figure 2 are remarkably stable when varying the number of groups. The
specification with G = 3 shows two groups essentially identical to groups 1 and 2 above, and a third
one that clusters groups 3 and 4 that experiences an upward trend in democracy over the period.
Allowing for G = 5 yields four groups similar to groups 1-4, plus another group whose democracy
level is intermediate between those of groups 1 and 2, roughly stable over time. This additional group
includes Mexico, Indonesia, and Turkey (12 countries in total). Table C8 in Appendix C shows group
membership by country, and Figure C3 the corresponding time patterns, for G = 2, ..., 6. When the
number of groups is 6 (or higher) the estimated group-specific time profiles tend to become more
volatile.
It is important to note that the time patterns and group membership reported in Figure 2 are
estimated from the data, and not driven by modelling assumptions other than the group structure.
In particular, nothing in our framework imposes that time patterns are smooth over time. Also,
group membership is not assumed to have a particular spatial structure, so the geographic correlation
apparent on the map is a result of estimation, not modelling assumptions. In particular, our approach
permits group membership and income– our main regressor– to be correlated, in addition to the direct
effect of income on democracy which we documented in Figure 1.
Although the estimated groups exhibit a strong spatial clustering, they do not match a simple
geographic division. To illustrate this, we report in Figure C4 in Appendix C the group-specific time
effects and averages of democracy and income, respectively, when the continents are used to form five
32
Figure 2: Patterns of heterogeneity, G = 4
high democracy low democracy early transition late transition
Time effects αgt Av. democracy Av. income
1970 1980 1990 2000
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
1970 1980 1990 20000
0.2
0.4
0.6
0.8
1
1970 1980 1990 20007
7.5
8
8.5
9
9.5
Note: See the notes to Figure 1. On the bottom panel, the left column reports the group-specific
time effects αgt. The other two columns show the group-specific averages of democracy and lagged
income, respectively. Calendar years (1970-2000) are shown on the x-axis. The top panel shows group
membership.
33
groups. The results show that, although this simple geographic division yields a clear separation in
terms of income and democracy levels, the time patterns are not as clearly separated as in Figure
2. In particular, this specification is not able to distinguish between stable and transition patterns
within South America or Africa. In contrast, the grouped fixed-effects estimator selects the grouping
that maximizes between-group variation, leading to a sharper identification of stable and transition
patterns.
As a different strategy, one could use external data to attempt to classify countries. This is the
approach taken by Papaioannou and Siourounis (2008), who combined electoral archives and historical
resources for this purpose. Interestingly, their classification of the type of political evolution closely
matches the results of GFE estimation.32 Note that, unlike this data-intensive alternative, our simple,
automatic method does not require the use of external data.
6.2.3 Robustness checks
We summarize the results of four sets of robustness checks.33 First, we use the unbalanced panel
that covers the period 1960-2000. After dropping all countries that have less than 3 observations,
we obtain an unbalanced sample of 118 countries.34 The cumulative income effect is close to the one
that we estimated on the balanced sample: for example, it is .13 for G = 4 and .12 for G = 10.
Interestingly, the group classification is very similar between the two samples: when G = 4 the group-
specific patterns also highlight high and low-democracy countries, as well as early and late transition
countries. Moreover, out of the 90 countries of the balanced sample, only 6 change groups when
estimated on the unbalanced panel.35
As a second check, we follow Acemoglu et al. (2008) and use a different measure of democracy:
the (normalized) composite Polity index. The balanced panel contains 75 countries, for the same time
periods. Grouped fixed-effects gives similar results to the ones using the Freedom House measure.
The income effect is .20 in the pooled OLS regression, .06 for GFE with G = 2, and decreases slightly
to .05 when G = 15, significant. Moreover, time patterns and country classification are also similar,
although there are some differences related to the measurement of democracy.36
32One of the few clear differences between their classification and ours is Iran, which is consistently classified as
a “low democracy” country according to our results, while Papaioannou and Siourounis classify it as a “borderline”
democratization case.33All the results that we refer to in this section, when not directly available in the text or appendix, are available from
the authors upon request.34The 32 countries we dropped using this selection criterion mostly belong to the ex-Republics of the Soviet Union,
which became independent in the second part of the sample.35All the countries whose group change switch from “late” to “early” transition. For example, Mexico, Philippines and
Taiwan become part of the early transition countries. As for countries that were not in the balanced sample: Haiti and
Zimbabwe are classified in group 2 (low democracy), Poland and Hungary in group 4 (late transition), and Botswana is
classified in group 1 (high-democracy).36For example, for G = 4 group membership coincides with the one shown in Table C8 in Appendix C except in
34
As a third check, we include additional controls in model (28). Specifically, following Acemoglu et
al. (2008) we control for education, log-population size, and age group percentages (5 categories, plus
median age). The results are very similar to the main specification. When controlling for education
and population size only, the income effect has a similar magnitude (≈ .10, significant), while when
adding age structure as a control the cumulative income effect drops to .05, marginally significant.
For both specifications the time patterns and classification documented in Figure 2 remain almost
unchanged.37
As a fourth and important check, we show the results of a model that combines time-varying
grouped-specific effects and time-invariant country-specific effects, as in equation (5). The model is
estimated using grouped fixed-effects in deviations to country-specific means. Table C6 in Appendix
C shows the estimates of the income effect. According to these results, the implied cumulative effect
of income on democracy is insignificant, in contrast with the quantitatively small but statistically
significant effect obtained using baseline GFE (see Figure 1). The income effect estimated using FE
and GFE at the same time is thus in line with the baseline fixed-effects estimate.
However, the estimated time patterns are remarkably robust to the inclusion of country FE. As
we discussed in Section 2, our approach allows to consistently estimate group membership even in
the presence of country-specific fixed-effects. The upper panel in Figure C5 in Appendix C shows
that a specification with G = 3 yields a similar division between “stable”, “early transition”, and
“late transition” countries. Moreover, the last column in Table C8 shows that the match with the
classification without country FE and G = 4 is perfect for 80 out of the 90 countries, the “stable”
group mostly comprising countries that belonged to groups 1 and 2 in the baseline specification (see
Figure 2). We also estimated the model without including lagged democracy as a control, in order
to alleviate potential concerns relative to the presence of the lagged outcome, and found very similar
results. Indeed, remarkably similar group classifications emerge when using the standard “kmeans”
algorithm (without covariates), both in levels and in deviations to country-specific means.
As a last exercise, we experimented with the two-layer model of unobserved heterogeneity (7).
This model has G1 groups with time-varying patterns, and each of these groups is divided into G2
subgroups whose time patterns differ from the common one by an intercept shift. The two-layer model
is more parsimonious than the one that combines GFE and FE, and may be well adapted given the
short length of the panel. We found it useful to allow for a different number of subgroups within each
group, and assume the following two-layer group structure:
(g1, g2) ∈ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 1), (2, 2), (3, 1), (3, 2) .11 cases. One of the major disagreements between the two sets of results is South Africa, which Polity classifies as a
democracy at the beginning of the period, while Freedom House classifies it as a non-democracy.37Interestingly, in both models that control for additional covariates, the BIC criterion selects G = 7 groups, a more
parsimonious and interpretable specification than in the case without additional covariates, see footnote 31.
35
The lower panel of Figure C5 in Appendix C shows the time-varying group-specific patterns, and
the next-to-last two columns in Table C8 show group membership by country. We see that the two-
layer model delivers a clear separation between stable countries, early transition countries, and late
transition countries. This output is similar to the baseline GFE specification with G = 4, and to
the estimates in deviations to country-specific means with G = 3. Note that the two-layer and GFE
specifications deliver almost identical group classifications (except in 5 cases).
In addition, the results provide evidence that the three time-varying groups are heterogeneous
themselves. Stable countries show the highest degree of heterogeneity, with 5 subgroups: high
democracy countries (such as the US, Japan, Western Europe), medium-high democracy (Colom-
bia, Venezuela), intermediate (Turkey, Malaysia), medium-low (Paraguay, Indonesia, Egypt), and low
democracy countries (China, Iran). Early transition countries are divided into high (Spain, Portugal)
and low (part of Latin America) democracy levels. Similarly, late transition countries are also divided
into high (South Africa, Panama) and low (part of Sub-saharian Africa). Note that the fact that stable
countries are separated into 5 subgroups, whereas early and late transition countries are divided into
2 subgroups each, is a result of estimation, not of modelling assumptions.
Overall, the evidence obtained suggests that the income effect is perhaps zero, or in any case
quantitatively small, in line with the conclusions of Acemoglu et al. (2008). At the same time, our
analysis highlights the presence of a strong clustering in the evolution of political outcomes: while a
substantial share of the world seems to have experienced stable parallel political patterns during the
period, roughly one third of the sample has seen a steep upward profile, at different points in time. In
the last part of this section we attempt to find an explanation for why these groups of countries have
evolved so differently.
6.2.4 Explaining the estimated grouped patterns
The country classification of Figure 2 seems to be a robust feature of the democracy/income relation-
ship data in the last third of the twentieth century. We now attempt to identify factors that explain
why these four estimated groups of countries are associated with such different levels and evolution of
democracy and income.
The first set of factors we consider are long-run, historical determinants. Following Acemoglu
et al. (2008), we consider a measure of constraints on the executive at independence, the rationale
being that more stringent constraints may be beneficial to embark on a pro-growth, pro-democracy
development path. We also consider the date of independence, and a measure of log GDP per capita
in 1500, as potential long-run determinants. In addition, we consider the initial democracy level (in
1965), as well as two factors that have been emphasized by the “modernization” theory (Lipset, 1959):
log-GDP per capita (in 1965), and a measure of education (average years of schooling, in 1970). We
also include shares of Catholic and Protestant in the population (in 1980).
36
Table C7 in Appendix C shows descriptive statistics by group. Both the high-democracy countries
(group 1) and the early transition ones (group 3) became independent in the nineteenth century on
average, while the countries in the two other groups became independent more recently. The high-
democracy group had more stringent constraints on the executive at the time of independence. This
group also has a higher initial democracy level in 1965,38 higher initial income and education, and a
larger share of Protestant. The early transition group (group 3) has a higher average education level
than the low democracy group, and a much larger share of Catholic (63% versus 23%). Lastly, the
late transition group (group 4) differs little from the low democracy one in terms of observables, apart
from a slightly higher education level.
In order to jointly assess the effects of the different factors, we next report in Table 5 the results of
multinomial logit regressions of the four estimated groups, using several specifications. The asymptotic
analysis of Section 4 provides a justification for treating the group estimates as data when running
the regressions and computing standard errors. The base category is group 2 (low-democracy).
The third row of Table 5 shows that constraints on the executive at independence is a significant
predictor of the probability of belonging to group 1 relative to group 2. This is consistent with the idea
that group 1 and group 2 countries have embarked on divergent paths at the time of independence, and
is suggestive of a very high persistence of early institutions. Notice that the effect remains significant
at the 10% level even when all other controls (democracy, income, education...) are included. At the
same time, early independence is also associated with a higher likelihood of belonging to group 1.
However, constraints on the executive at independence do not significantly affect the probability of
belonging to either of the two transition groups (3 and 4). This suggests that, while long-run, historical
factors partly explain differences between stable (low versus high-democracy) countries, they are not
able to explain the remarkable evolution of transition countries during the recent period.
Factors that affect the probability to belong to the early transition group are the independence
date– of difficult interpretation– and, in line with the “modernization” theory, the education level.39
In contrast, the table shows that none of the determinants that we consider is able to distinguish
late transition countries (group 4) from low democracy countries (group 2). In particular, neither
education nor religion have significant coefficients.
Our framework, which is useful to estimate “wave” patterns of democratization, sheds some light
on the difficult problem of identifying the causes of the waves. In fact, our results leave many questions
unanswered. Which factors explain democratic transitions? Why did a large share of low democracy
38Note that the group averages of democracy in 1965 are higher for groups 2-4 than the 1970 levels that can be seen
on Figure 2. This reflects the fact that the 1960s were characterized by a number of transitions to autocracy, a feature
that we also observed on our estimates from the 1960− 2000 unbalanced sample.39These results are consistent with Papaioannou and Siourounis (2008), who modelled the probability of democratiza-
tion of countries that started the period as autocracies. They found little evidence for an effect of early institutions. In
addition, their results suggest that more educated societies are more likely to become democratic.
37
Table 5: Explaining group membership
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
Group 1: high-democracy (vs group 2: low democracy)
log GDP p.c. (1500) 1.39(.971)
.865(1.74)
.698(1.76)
− − − −.224(2.41)
−.307(2.61)
−.465(2.75)
−.628(2.67)
Independence Year/100 − −4.55(1.22)
−4.44(1.27)
− − − −3.51(1.43)
−3.72(1.56)
−3.68(1.75)
−3.59(1.75)
Constraints − 7.26(2.00)
7.12(2.06)
− − − 5.67(2.49)
4.74(2.60)
4.70(2.62)
4.52(2.77)
Democracy (1965) − − − 7.10(2.11)
5.80(2.56)
5.92(2.66)
− 6.72(3.39)
6.81(3.44)
6.24(3.65)
log GDP p.c. (1965) − − − 1.51(.587)
− 1.09(.883)
− − .194(1.25)
.447(1.35)
Education (1970) − − − − .798(.324)
.492(.402)
.949(.373)
.443(.435)
.418(.536)
.258(.560)
Share Catholic (1980) − − .611(1.20)
− − − − − − −.627(1.70)
Share Protestant (1980) − − 6.81(4.37)
− − − − − − 3.85(6.32)
Group 3: early transition (vs group 2: low democracy)
log GDP p.c. (1500) .959(1.19)
−.894(1.85)
−.504(1.87)
− − − −1.19(2.44)
−2.27(2.56)
−3.48(3.03)
−3.13(2.97)
Independence Year/100 − −3.53(1.11)
−3.32(1.23)
− − − −2.72(1.23)
−2.96(1.30)
−4.02(1.63)
−3.82(1.76)
Constraints − 2.25(2.10)
2.23(2.34)
− − − .939(2.47)
.473(2.56)
.070(2.57)
.010(2.95)
Democracy (1965) − − − −.232(1.69)
−1.63(2.03)
−1.79(2.08)
− −1.36(3.03)
−.831(3.02)
−1.37(3.16)
log GDP p.c. (1965) − − − 1.40(.567)
− .503(.793)
− − −1.87(1.34)
−1.58(1.42)
Education (1970) − − − − .883(.311)
.749(.357)
.570(.361)
.729(.425)
1.19(.565)
1.18(.601)
Share Catholic (1980) − − 1.00(1.22)
− − − − − − −.215(1.67)
Share Protestant (1980) − − −.552(7.87)
− − − − − − −1.55(8.93)
Group 4: late transition (vs group 2: low democracy)
log GDP p.c. (1500) −1.06(1.14)
−.968(1.07)
−.751(1.14)
− − − −1.63(1.95)
−1.97(2.07)
−1.99(2.13)
−2.08(2.16)
Independence Year/100 − −.681(.635)
−.785(.763)
− − − −.027(.926)
−.144(.939)
−.219(1.03)
−.007(1.38)
Constraints − .485(1.30)
.848(1.39)
− − − −.607(1.74)
−1.05(1.86)
−1.11(1.88)
−.527(2.22)
Democracy (1965) − − − 1.23(1.43)
.047(1.93)
.134(1.89)
− 2.39(2.45)
2.46(2.45)
1.50(2.77)
log GDP p.c. (1965) − − − .021(.464)
− −.215(.701)
− − −.263(.902)
.214(1.07)
Education (1970) − − − − .494(.302)
.544(.349)
.597(.358)
.423(.389)
.502(.439)
.331(.476)
Share Catholic (1980) − − .888(1.19)
− − − − − − 1.20(1.90)
Share Protestant (1980) − − 5.40(3.87)
− − − − − − 5.23(5.78)
Note: Balanced panel from Acemoglu et al. (2008). “Constraints” are constraints on the executive at independence,
measured as in Acemoglu et al. (2005). Multinomial logit regressions of the estimated groups (G = 4). The reference
group is group 2 (low-democracy). Group membership is shown on Figure 2. Sample size in the most flexible specification–
column (10)– is N = 68.38
countries– including a substantial share of Africa– make a transition in the 1990s?40 Lastly, and
importantly, why do we observe groups of countries making transitions at similar points in time?
7 Conclusion
Grouped fixed-effects (GFE) offers a flexible yet parsimonious approach to model unobserved hetero-
geneity patterns. The approach delivers estimates of common regression parameters, together with
interpretable estimates of group-specific time patterns and group membership. The framework allows
for strictly exogenous or predetermined covariates, and can allow for unit-specific fixed-effects in addi-
tion to the time-varying grouped patterns. Importantly, the relationship between group membership
and observables is left unrestricted. A priori information– when available– may be incorporated in a
simple way.
The GFE approach should be useful in applications where time-invariance of the fixed-effects
is a problematic assumption, and where time-varying grouped effects may be present in the data.
As a first example, the empirical analysis of the evolution of democracy shows clear evidence of a
clustering of political regimes and transitions. More generally, GFE should be well-suited in difference-
in-difference designs, as a way to relax parallel trend assumptions. Other potential applications include
social interactions models where reference groups are estimated from the data, and models of spatial
dependence with an endogenous spatial weights matrix. Computation of the estimator is challenging
but not impossible, thanks to recent advances in the literature on data clustering. Assessing how well
our algorithm performs in larger datasets than the one we have used is certainly important for future
applications.
Our asymptotic results show that, though subject to an incidental parameter problem, GFE has
attractive large-N,T properties. In particular, there is no need to perform (higher-order) bias reduc-
tion. However, two issues seems worth studying in this context. First, the main asymptotic equivalence
result relies on groups being well separated. This assumption may be a strong one. For example, sim-
ulation experiments that we have performed suggest that group separation is more likely to fail in
models where unobserved heterogeneity is time-invariant. Also, group separation is clearly violated
when the postulated number of groups exceeds the true one. Providing asymptotic results that hold
uniformly with respect to the values of the group-specific parameters seems an interesting research
avenue.
A second interesting extension of the asymptotic analysis concerns letting the number of groups
G grow with the sample size. While G is kept fixed in our main theoretical results, our simulations
40Note that most of the late transition countries are Sub-saharian African countries, which became (more) democratic in
the 1990s. Interestingly, Bruckner and Ciccone (2011) document an association between drought and posterior increases
in democracy levels in Sub-saharian Africa. They interpret this evidence as suggesting that a fall in transitory income
may foster democratic change.
39
suggest that common parameter estimates may still behave well in situations where G is actually larger
than the number of time periods. Bester and Hansen (2010) conduct an analysis where G tends to
infinity with both dimensions of the panel, and emphasize a trade-off between misspecification bias
and incidental parameter bias. It seems worth studying this trade-off in a context where the data
grouping is unknown and needs to be estimated.
Lastly, an important research question is the study of the GFE approach in nonlinear models. We
are particularly interested in dynamic discrete choice models, where a discrete modelling of unobserved
heterogeneity may be appealing (Kasahara and Shimotsu, 2009, Browning and Carro, 2011). The
computation and statistical properties of our approach in these models raises specific challenges, which
we plan to address in future work.
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45
APPENDIX
A Proofs
A.1 Proof of Theorem 1
Let γ0 = g01 , ..., g0N denote the population grouping. Let also γ = g1, ..., gN denote any grouping of the
cross-sectional units into G groups.
Let us define:
Q (θ, α, γ) =1
NT
N∑
i=1
T∑
t=1
(yit − x′itθ − αgit)2. (A1)
Note that the GFE estimator minimizes Q (·) over all (θ, α, γ) ∈ Θ×AGT × ΓG. Note also that:
Q (θ, α, γ) =1
NT
N∑
i=1
T∑
t=1
(vit + x′it
(θ0 − θ
)+ α0
g0i t− αgit
)2.
We also define the following auxiliary objective function:
Q (θ, α, γ) =1
NT
N∑
i=1
T∑
t=1
(x′it(θ0 − θ
)+ α0
g0i t− αgit
)2+
1
NT
N∑
i=1
T∑
t=1
v2it.
We start by showing the following uniform convergence result.
Lemma A1 Let Assumption 1.a-1.g hold. Then:
plimN,T→∞
sup(θ,α,γ)∈Θ×AGT×ΓG
∣∣∣Q (θ, α, γ)− Q (θ, α, γ)∣∣∣ = 0.
Proof.
Q (θ, α, γ)− Q (θ, α, γ) =2
NT
N∑
i=1
T∑
t=1
vit
(x′it(θ0 − θ
)+ α0
g0i t− αgit
)
=
(2
NT
N∑
i=1
T∑
t=1
vitxit
)′ (θ0 − θ
)+
2
NT
N∑
i=1
T∑
t=1
vitα0g0i t
− 2
NT
N∑
i=1
T∑
t=1
vitαgit.
We now show that the three terms on the right-hand side of this equality are op(1), uniformly on the
parameter space.
• By Assumption 1.e we have:
E
1
N
N∑
i=1
∥∥∥∥∥1
T
T∑
t=1
vitxit
∥∥∥∥∥
2 ≤ M
T,
so it follows from the Cauchy-Schwartz (CS) inequality that 2NT
∑Ni=1
∑Tt=1 vitxit = op(1), uniformly on the
parameter space. In addition,∥∥θ0 − θ
∥∥ is bounded by Assumption 1.a.
46
• By the CS inequality:
(1
NT
N∑
i=1
T∑
t=1
vitα0g0i t
)2
≤ 1
N
N∑
i=1
(1
T
T∑
t=1
vitα0g0i t
)2
=
G∑
g=1
1
N
N∑
i=1
1g0i = g
(
1
T
T∑
t=1
vitα0gt
)2
≤G∑
g=1
1
N
N∑
i=1
(1
T
T∑
t=1
vitα0gt
)2
=G∑
g=1
1
NT 2
N∑
i=1
T∑
t=1
T∑
s=1
vitvisα0gtα
0gs,
which by Assumption 1.d is bounded in expectation by a constant divided by T . This implies that 1NT
∑Ni=1
∑Tt=1 vitα
0g0i t
is uniformly op(1).
• Finally we have:
1
NT
N∑
i=1
T∑
t=1
vitαgit =
G∑
g=1
[1
NT
N∑
i=1
T∑
t=1
1gi = gvitαgt
]
=
G∑
g=1
[1
T
T∑
t=1
αgt
(1
N
N∑
i=1
1gi = gvit)]
.
Moreover, by the CS inequality and for all g ∈ 1, ..., G:(
1
T
T∑
t=1
αgt
(1
N
N∑
i=1
1gi = gvit))2
≤(
1
T
T∑
t=1
α2gt
)×
1
T
T∑
t=1
(1
N
N∑
i=1
1gi = gvit)2 ,
where, by Assumption 1.a, 1T
∑Tt=1 α
2gt is bounded uniformly.
Now, note that:
1
T
T∑
t=1
(1
N
N∑
i=1
1gi = gvit)2
=1
TN2
N∑
i=1
N∑
j=1
1gi = g1gj = gT∑
t=1
vitvjt
≤ 1
N2
N∑
i=1
N∑
j=1
∣∣∣∣∣1
T
T∑
t=1
vitvjt
∣∣∣∣∣
≤ 1
N2
N∑
i=1
N∑
j=1
∣∣∣∣∣1
T
T∑
t=1
E (vitvjt)
∣∣∣∣∣
+1
N2
N∑
i=1
N∑
j=1
∣∣∣∣∣1
T
T∑
t=1
(vitvjt − E (vitvjt))
∣∣∣∣∣ .
By Assumption 1.f:
1
N2
N∑
i=1
N∑
j=1
∣∣∣∣∣1
T
T∑
t=1
E (vitvjt)
∣∣∣∣∣ ≤M
N.
By the CS inequality: 1
N2
N∑
i=1
N∑
j=1
∣∣∣∣∣1
T
T∑
t=1
(vitvjt − E (vitvjt))
∣∣∣∣∣
2
≤ 1
N2
N∑
i=1
N∑
j=1
(1
T
T∑
t=1
(vitvjt − E (vitvjt))
)2
,
47
which is bounded in expectation by M/T by Assumption 1.g.
This shows that 1NT
∑Ni=1
∑Tt=1 vitαgit is uniformly op(1), and ends the proof of Lemma A1.
We will also need the following result, which shows that Q (·) is maximized at true values.
Lemma A2 We have, for all (θ, α, γ) ∈ Θ×AGT × ΓG:
Q (θ, α, γ)− Q(θ0, α0, γ0
)≥ ρ
∥∥θ − θ0∥∥2 ,
where ρ is given by Assumption 1.h.
Proof. Let us denote, for every grouping γ = g1, ..., gN:
Σ (γ) =1
NT
N∑
i=1
T∑
t=1
(xit − xg0
i∧gi,t
)(xit − xg0
i∧gi,t
)′.
We have, from standard least-squares algebra:
Q (θ, α, γ)− Q(θ0, α0, γ0
)=
1
NT
N∑
i=1
T∑
t=1
(x′it(θ0 − θ
)+ α0
g0i t− αgit
)2
≥(θ0 − θ
)′Σ(γ)
(θ0 − θ
)
≥(θ0 − θ
)′(inf
γ∈ΓG
Σ(γ)
)(θ0 − θ
)
≥ ρ∥∥θ0 − θ
∥∥2 ,
where ρ is given by Assumption 1.h.
To show that θ is consistent for θ0, note that, by Lemma A1 and by the definition of the GFE estimator we
have:
Q(θ, α, γ
)= Q
(θ, α, γ
)+ op(1)
≤ Q(θ0, α0, γ0
)+ op(1)
= Q(θ0, α0, γ0
)+ op(1). (A2)
So, by Lemma A2 we have:
ρ∥∥∥θ − θ0
∥∥∥2
= op(1),
so it follows from Assumption 1.h that∥∥∥θ − θ0
∥∥∥2
= op(1).
Lastly, to show convergence in quadratic mean of the estimated unit-specific effects note that, by the CS
inequality:
∣∣∣Q(θ, α, γ
)− Q
(θ0, α, γ
)∣∣∣ =
∣∣∣∣∣1
NT
N∑
i=1
T∑
t=1
x′it(θ0 − θ
) [x′it(θ0 − θ
)+ 2
(α0g0i t− αgit
)]∣∣∣∣∣
≤ 1
NT
N∑
i=1
T∑
t=1
‖xit‖2 ×∥∥∥θ0 − θ
∥∥∥2
+
(4 supαt∈A
|αt|)× 1
NT
N∑
i=1
T∑
t=1
‖xit‖ ×∥∥∥θ0 − θ
∥∥∥ ,
48
which is op(1) by Assumptions 1.a and 1.b, and by consistency of θ.
Combining with (A2) we obtain:
Q(θ0, α, γ
)≤ Q
(θ0, α0, γ0
)+ op(1),
from which it follows that:
1
NT
N∑
i=1
T∑
t=1
(αgit − α0
g0i t
)2= op(1).
This completes the proof of Theorem 1.
A.2 Proof of Theorem 2
We first establish that α is consistent for α0. Because the objective function is invariant to re-labelling of the
groups, we show consistency with respect to the Hausdorff distance:
dH (a, b) = max
max
g∈1,...,G
(min
g∈1,...,G
1
T
T∑
t=1
(agt − bgt)2
), maxg∈1,...,G
(min
g∈1,...,G
1
T
T∑
t=1
(agt − bgt)2
).
We have the following result.41
Lemma A3 Let Assumptions 1.a-1.h, and 2.a-2.b hold. Then, as N and T tend to infinity:
dH(α, α0
) p→ 0.
Proof.
We study the two terms in the max·, · in turn.
• Let g ∈ 1, ..., G. We have:
1
NT
N∑
i=1
(min
g∈1,...,G
T∑
t=1
1g0i = g(αgt − α0
gt
)2)
=
(1
N
N∑
i=1
1g0i = g)
× ...
(min
g∈1,...,G1
T
T∑
t=1
(αgt − α0
gt
)2).
By Assumption 2.a it is thus enough to show that, for all g, as N and T tend to infinity:
1
NT
N∑
i=1
(min
g∈1,...,G
T∑
t=1
1g0i = g(αgt − α0
gt
)2)
p→ 0.
Now:
1
NT
N∑
i=1
(min
g∈1,...,G
T∑
t=1
1g0i = g(αgt − α0
gt
)2)
≤ 1
NT
N∑
i=1
T∑
t=1
1g0i = g(αgit − α0
g0i t
)2
≤ 1
NT
N∑
i=1
T∑
t=1
(αgit − α0
g0i t
)2,
41Note that group separation (Assumption 2.b) is assumed to derive the result. Proving consistency of the group-
specific time effects absent this assumption would require different arguments.
49
which tends to zero in probability by Theorem 1.
We have thus shown that, for all g:
ming∈1,...,G
1
T
T∑
t=1
(αgt − α0
gt
)2 p→ 0. (A3)
• It follows from (A3) and the fact that 1, ..., G is finite that there exists some mapping σ : 1, ..., G →1, ..., G such that, for all g and for all ε > 0 we have, with probability approaching one:
1
T
T∑
t=1
(ασ(g)t − α0
gt
)2< ε. (A4)
We now show that σ(·) is one-to-one.Let g 6= g. By the triangular inequality we have, using (A4) twice (at g and g):
(1
T
T∑
t=1
(ασ(g)t − ασ(g)t
)2) 1
2
≥(
1
T
T∑
t=1
(α0gt − α0
gt
)2) 1
2
− 2ε,
where 1T
∑Tt=1
(α0gt − α0
gt
)2is bounded from below as T tends to infinity by Assumption 2.b. So, by choosing ε
small enough it follows that σ(g) 6= σ(g). This implies that σ : 1, ..., G → 1, ..., G is one-to-one and admits
a well-defined inverse σ−1.
Finally, it thus follows that, for all g ∈ 1, ..., G and as T tends to infinity:
ming∈1,...,G
1
T
T∑
t=1
(αgt − α0
gt
)2 ≤ 1
T
T∑
t=1
(αgt − α0
σ−1(g)t
)2 p→ 0,
where we have used (A4) with g = σ−1(g).
This completes the proof.
The proof of Lemma A3 shows that there exists a permutation σ : 1, ..., G → 1, ..., G such that:
1
T
T∑
t=1
(ασ(g)t − α0
gt
)2 p→ 0.
By simple relabelling of the elements of α we may take σ(g) = g. We adopt this convention in the rest of the
proof.
For any η > 0, let Nη denote the set of parameters (θ, α) ∈ Θ × AGT that satisfy∥∥θ − θ0
∥∥2 < η and1T
∑Tt=1
(αgt − α0
gt
)2< η for all g ∈ 1, ..., G. We have the following result, which shows that the probability
that gi (θ, α) differs from g0i tends to zero at a faster-than-polynomial rate, provided (θ, α) is taken in a small
enough neighborhood Nη.
Lemma A4 For η > 0 small enough we have, for all δ > 0 and as T tends to infinity:
sup(θ,α)∈Nη
1
N
N∑
i=1
1gi (θ, α) 6= g0i = op(T−δ
).
50
Proof.
Note that, from the definition of gi(·) we have, for all g ∈ 1, ..., G:
1gi (θ, α) = g ≤ 1
T∑
t=1
(yit − x′itθ − αgt)2 ≤
T∑
t=1
(yit − x′itθ − αg0
i t
)2.
So:
1
N
N∑
i=1
1gi (θ, α) 6= g0i =
G∑
g=1
1
N
N∑
i=1
1g0i 6= g1gi (θ, α) = g
≤G∑
g=1
1
N
N∑
i=1
1g0i 6= g1
T∑
t=1
(yit − x′itθ − αgt)2 ≤
T∑
t=1
(yit − x′itθ − αg0
i t
)2
︸ ︷︷ ︸Zig(θ,α)
.
(A5)
We start by bounding Zig(θ, α), for all (θ, α) ∈ Nη, by a quantity that does not depend on (θ, α). To
proceed note that, for all (θ, α) and all i:
Zig(θ, α) ≤ maxg 6=g
1
T∑
t=1
(yit − x′itθ − αgt)2 ≤
T∑
t=1
(yit − x′itθ − αgt)2
= maxg 6=g
1
T∑
t=1
(αgt − αgt)
(yit − x′itθ −
αgt + αgt
2
)≤ 0
= maxg 6=g
1
T∑
t=1
(αgt − αgt)
(vit + x′it
(θ0 − θ
)+ α0
gt −αgt + αgt
2
)≤ 0
.
Let us now define:
AT =
∣∣∣∣∣T∑
t=1
(αgt − αgt)
(vit + x′it
(θ0 − θ
)+ α0
gt −αgt + αgt
2
)−
T∑
t=1
(α0gt − α0
gt
)(vit + α0
gt −α0gt + α0
gt
2
)∣∣∣∣∣ .
We have:
AT ≤∣∣∣∣∣
T∑
t=1
(αgt − αgt) vit −T∑
t=1
(α0gt − α0
gt
)vit
∣∣∣∣∣︸ ︷︷ ︸
=A1T
+
∣∣∣∣∣T∑
t=1
(αgt − αgt)x′it
(θ0 − θ
)∣∣∣∣∣
︸ ︷︷ ︸=A2T
+
∣∣∣∣∣T∑
t=1
(αgt − αgt)
(α0gt −
αgt + αgt
2
)−
T∑
t=1
(α0gt − α0
gt
)(α0gt −
α0gt + α0
gt
2
)∣∣∣∣∣︸ ︷︷ ︸
=A3T
.
We now bound each of the three terms, for (θ, α) ∈ Nη.
• We have, by the Cauchy-Schwartz inequality:
A1T =
∣∣∣∣∣T∑
t=1
[(αgt − αgt)−
(α0gt − α0
gt
)]vit
∣∣∣∣∣
≤ T
(1
T
T∑
t=1
[(αgt − αgt)−
(α0gt − α0
gt
)]2) 1
2
×(
1
T
T∑
t=1
v2it
) 12
≤ TC1√η
(1
T
T∑
t=1
v2it
) 12
,
51
where C1 is independent of η and T , and where we have used the definition of Nη.
• Next we have, by the CS inequality:
A2T =
∣∣∣∣∣T∑
t=1
(αgt − αgt)x′it
(θ0 − θ
)∣∣∣∣∣
≤ T
(2 supαt∈A
|αt|)×(
1
T
T∑
t=1
‖xit‖)
×∥∥θ0 − θ
∥∥
≤ TC2√η
(1
T
T∑
t=1
‖xit‖),
where C2 is independent of η and T , and where we have used Assumption 1.a.
• Finally we have, by simple rearrangement:
A3T =
∣∣∣∣∣T∑
t=1
(αgt − αgt)
(α0gt −
αgt + αgt
2
)−
T∑
t=1
(α0gt − α0
gt
)(α0gt −
α0gt + α0
gt
2
)∣∣∣∣∣
=
∣∣∣∣∣T∑
t=1
α0gt
(αgt − α0
gt − αgt + α0gt
)+
1
2
T∑
t=1
([α0gt
]2 − [αgt]2 −
[α0gt
]2+ [αgt]
2)∣∣∣∣∣ .
It thus follows from the CS inequality and Assumption 1.a that, for (θ, α) ∈ Nη:
A3T ≤ TC3√η,
where C3 is independent of η and T .
Combining, we obtain that:
Zig(θ, α) ≤ maxg 6=g
1
T∑
t=1
(αgt − αgt)
(vit + x′it
(θ0 − θ
)+ α0
gt −αgt + αgt
2
)≤ 0
≤ maxg 6=g
1
T∑
t=1
(α0gt − α0
gt
)(vit + α0
gt −α0gt + α0
gt
2
)≤ AT
≤ maxg 6=g
1
T∑
t=1
(α0gt − α0
gt
)(vit + α0
gt −α0gt + α0
gt
2
)
≤ TC1√η
(1
T
T∑
t=1
v2it
) 12
+ TC2√η
(1
T
T∑
t=1
‖xit‖)
+ TC3√η
.
Note also that:
T∑
t=1
(α0gt − α0
gt
)(α0gt −
α0gt + α0
gt
2
)=
1
2
T∑
t=1
(α0gt − α0
gt
)2
≥ Tc
2,
where we have used Assumption 2.b.
Hence we have:
Zig(θ, α) ≤ Zig,
52
where:
Zig = maxg 6=g
1
T∑
t=1
(α0gt − α0
gt
)vit ≤ −T c
2+ TC1
√η
(1
T
T∑
t=1
v2it
) 12
+TC2√η
(1
T
T∑
t=1
‖xit‖)
+ TC3√η
.
Note that Zig does not depend on (θ, α). In particular we also have:
sup(θ,α)∈Nη
Zig(θ, α) ≤ Zig,
and thus:
sup(θ,α)∈Nη
1
N
N∑
i=1
1gi (θ, α) 6= g0i ≤ 1
N
N∑
i=1
G∑
g=1
Zig. (A6)
Fix M >√M , whereM is given by Assumption 1. Note that E(v2it) ≤
√M , and E(‖xit‖) ≤
√M . We have,
using standard probability algebra and for all g:
Pr(Zig = 1
)≤
∑
g 6=g
Pr
(T∑
t=1
(α0gt − α0
gt
)vit ≤ −T c
2+ TC1
√η
(1
T
T∑
t=1
v2it
) 12
+TC2√η
(1
T
T∑
t=1
‖xit‖)
+ TC3√η
)
≤∑
g 6=g
[Pr
(1
T
T∑
t=1
v2it ≥ M
)+ Pr
(1
T
T∑
t=1
‖xit‖ ≥ M
)
+Pr
(T∑
t=1
(α0gt − α0
gt
)vit ≤ −T c
2+ TC1
√η√M
+TC2√ηM + TC3
√η
)].
(A7)
To end the proof of Lemma A4, we rely on the use of exponential inequalities for dependent processes.
Specifically, we use the following result, due to Rio (2000, Theorem 6.2) and expressed in this form by Merlevede,
Peligrad and Rio (2009).
Theorem 3 (Rio) Let zt be a strongly mixing process with zero mean, with strong mixing coefficients α[t] ≤e−atd1 , and with tail probabilities Pr(|zt| > z) ≤ e1−(
zb )
d2
, where a, b, d1, and d2 are positive constants. Let
s2 = 1T
∑Tt=1
∑Ts=1 E (ztzs) <∞, and let d = d1d2
d1+d2. Then there exists a constant f > 0 independent of T such
that, for all z and T :
Pr
(∣∣∣∣∣1
T
T∑
t=1
zt
∣∣∣∣∣ ≥ z
)≤ 4
(1 + T
32z2
16s2
)− 12T
12
+16f
ze−a
(T
12 z
4b
)d
.
Proof. Evaluate inequality (1.7) in Merlevede, Peligrad and Rio (2009) at λ = T z4 , and take r = T
12 .
We now bound the three terms on the right-hand side of (A7).
53
• Applying Rio’s theorem to zt = v2it − E(v2it) and taking z = M −√M yields:
Pr
(1
T
T∑
t=1
v2it ≥ M
)= o
(T−δ
)
for all δ > 0. Note that v2itt is strongly mixing as vitt is strongly mixing by Assumption 2.c.
• As for the second term there are two cases, depending on whether part (i) or (ii) is satisfied in Assumption
2.e. If part (i) holds then by choosing M larger than the upper bound on ‖xit‖ yields 1T
∑Tt=1 ‖xit‖ < M .
If part (ii) in Assumption 2.e holds then we apply Rio’s theorem to zt = ‖xit‖ − E(‖xit‖) and take z =
M −√M , yielding:
Pr
(1
T
T∑
t=1
‖xit‖ ≥ M
)= o
(T−δ
).
• Lastly, to bound the third term in (A7) we take:
η ≤
c
4(C1
√M + C2M + C3
)
2
. (A8)
Note that the upper bound on η does not depend on T .
Taking η satisfying (A8) yields, for all g 6= g:
Pr
(1
T
T∑
t=1
(α0gt − α0
gt
)vit ≤ − c
2+ C1
√η√M + C2
√ηM + C3
√η
)≤ Pr
(1
T
T∑
t=1
(α0gt − α0
gt
)vit ≤ − c
4
).
Now, by Assumption 2.c the process(α0gt − α0
gt
)vit
tis strongly mixing with faster-than-polynomial
decay rate. Moreover, for all i, t, and m > 0:
Pr(∣∣(α0
gt − α0gt
)vit∣∣ > m
)≤ Pr
|vit| >
m
2 supαt∈A
|αt|
,
so(α0gt − α0
gt
)vit
talso satisfies the tail condition of Assumption 2.d, albeit with a different constant b > 0
instead of b > 0.
Lastly, applying Rio’s theorem with zt =(α0gt − α0
gt
)vit and taking z = c
4 yields:
Pr
(1
T
T∑
t=1
(α0gt − α0
gt
)vit ≤ − c
4
)= o
(T−δ
).
Combining the results we finally obtain, using (A7), that for η satisfying (A8) and for all δ > 0:
Pr(Zig = 1
)= o
(T−δ
).
Moreover, noting that the above upper bounds on the probabilities do not depend on i and g we have:
supi∈1,...,N,g∈1,...,G
Pr(Zig = 1
)= o
(T−δ
). (A9)
54
To complete the proof of Lemma A4 note that, for η that satisfies (A8) we have, for all δ > 0 and all ε > 0:
Pr
(sup
(θ,α)∈Nη
1
N
N∑
i=1
1gi (θ, α) 6= g0i > εT−δ
)≤ Pr
(1
N
N∑
i=1
G∑
g=1
Zig > εT−δ
)
≤E
(1N
∑Ni=1
∑Gg=1 Zig
)
εT−δ
≤ G
εT−δ×(
supi∈1,...,N,g∈1,...,G
Pr(Zig = 1
))
= o(1),
where we have used (A6), the Markov inequality, and (A9), respectively.
This ends the proof of Lemma A4.
We now prove the three parts of Theorem 2, relative to θ, α, and gi, in turn.
Properties of θ. Let us denote:
Q (θ, α) =1
NT
N∑
i=1
T∑
t=1
(yit − x′itθ − αgi(θ,α)t
)2, (A10)
and:
Q (θ, α) =1
NT
N∑
i=1
T∑
t=1
(yit − x′itθ − αg0
i t
)2. (A11)
Note that Q(·) is minimized at(θ, α
), and that Q(·) is minimized at
(θ, α
).
By the CS inequality we have:
(Q (θ, α)− Q (θ, α)
)2≤ 2
NT
N∑
i=1
T∑
t=1
(αg0
i t− αgi(θ,α)t
)2× ...
2
NT
N∑
i=1
T∑
t=1
(yit − x′itθ −
αgi(θ,α)t + αg0i t
2
)2
,
where the second term on the right-hand side is uniformly Op(1) by Assumptions 1.a-1.c.
Now we have:
1
NT
N∑
i=1
T∑
t=1
(αg0
i t− αgi(θ,α)t
)2=
1
NT
N∑
i=1
T∑
t=1
1gi (θ, α) 6= g0i (αg0
i t− αgi(θ,α)t
)2
≤(4 supαt∈A
α2t
)× 1
N
N∑
i=1
1gi (θ, α) 6= g0i .
Let η > 0 be small enough such that Lemma A4 is satisfied. Using the two above inequalities, Assumption
1.a, and Lemma A4 we have, for all δ > 0:
sup(θ,α)∈Nη
∣∣∣Q (θ, α)− Q (θ, α)∣∣∣ = op
(T−δ
).
55
Now, by consistency of θ (Theorem 1) and α (Lemma A3) we have, as N and T tend to infinity:
Pr((θ, α
)/∈ Nη
)N,T→∞→ 0.
Likewise, as θ and α are also consistent under the conditions of Theorem 1 we have:
Pr((θ, α
)/∈ Nη
)N,T→∞→ 0.
We thus have, as N and T tend to infinity:
Q(θ, α
)− Q
(θ, α
)= op
(T−δ
), (A12)
and:
Q(θ, α
)− Q
(θ, α
)= op
(T−δ
).
Next, note that, by the definition of(θ, α
):
Q(θ, α
)− Q
(θ, α
)≥ 0.
Moreover, using the above and the definition of(θ, a):
Q(θ, α
)− Q
(θ, α
)= Q
(θ, α
)− Q
(θ, α
)+ op(T
−δ)
≤ op(T−δ
).
It thus follows that:
Q(θ, α
)− Q
(θ, α
)= op
(T−δ
). (A13)
Now, we have:
Q(θ, α
)− Q
(θ, α
)=
2
NT
N∑
i=1
T∑
t=1
(x′it(θ − θ
)+ αg0
i t− αg0
i t
)(yit − x′it
(θ + θ
2
)−αg0
i t+ αg0
i t
2
)
=(θ − θ
)′ 2
NT
N∑
i=1
T∑
t=1
xit
(yit − x′it
(θ + θ
2
)−αg0
i t+ αg0
i t
2
)
+1
T
G∑
g=1
T∑
t=1
(αgt − αgt)2
N
N∑
i=1
1g0i = g(yit − x′it
(θ + θ
2
)− αgt + αgt
2
).
(A14)
Note that, as(θ, a)is a least squares estimator, the following empirical moment restrictions are satisfied:
2
NT
N∑
i=1
T∑
t=1
xit
(yit − x′itθ − αg0
i t
)= 0
2
N
N∑
i=1
1g0i = g(yit − x′itθ − αgt
)= 0, for all (g, t).
56
Combining with (A14) yields:
Q(θ, α
)− Q
(θ, α
)=
(θ − θ
)′ 2
NT
N∑
i=1
T∑
t=1
xit
(x′it
(θ − θ
2
)+
G∑
g=1
1g0i = g(αgt − αgt
2
))
+1
T
G∑
g=1
T∑
t=1
(αgt − αgt)2
N
N∑
i=1
1g0i = g(x′it
(θ − θ
2
)+αgt − αgt
2
)
(A15)
=1
NT
N∑
i=1
T∑
t=1
(x′it(θ − θ
)+
G∑
g=1
1g0i = g (αgt − αgt)
)2
,
so that:
Q(θ, α
)− Q
(θ, α
)≥
(θ − θ
)′(
1
NT
N∑
i=1
T∑
t=1
(xit − xg0
i ,t
)(xit − xg0
i ,t
)′)(
θ − θ).
It thus follows that:
Q(θ, α
)− Q
(θ, α
)≥ ρ
∥∥∥θ − θ∥∥∥2
,
where ρp→ ρ > 0 as a consequence of Assumption 1.h.
Hence, θ − θ = op(T−δ
)for all δ > 0.
Properties of α. Using (A15) above, consistency of θ and θ, Assumptions 1.a and 1.b, and equation (A13),
we obtain:
1
T
G∑
g=1
T∑
t=1
(αgt − αgt)2
N
N∑
i=1
1g0i = g(αgt − αgt
2
)= op
(T−δ
).
Using Assumption 2.a we thus have, for all g:
1
T
T∑
t=1
(αgt − αgt)2
= op(T−δ
).
In particular, for all t we have: (αgt − αgt)2 ≤ op
(T 1−δ
). As this holds for all δ > 0 we obtain the desired
result.
Properties of gi = gi
(θ, α
). Finally, we have:
Pr
(sup
i∈1,...,N
∣∣∣gi(θ, α
)− g0i
∣∣∣ > 0
)≤ Pr
((θ, α
)/∈ Nη
)+ E
[sup
(θ,α)∈Nη
Pr
(sup
i∈1,...,N
∣∣gi (θ, α)− g0i∣∣ > 0
)].
Note that the neighborhood Nη depends on the processes α0gtt, for g = 1, ..., G. This explains the presence of
an expectation on the right-hand side of this inequality.
Now we have, taking η such that (A8) is satisfied:
Pr((θ, a)/∈ Nη
)= o(1).
57
We also have:
sup(θ,α)∈Nη
Pr
(sup
i∈1,...,N
∣∣gi (θ, α)− g0i∣∣ > 0
)≤ N sup
(θ,α)∈Nη
supi∈1,...,N
Pr(∣∣gi (θ, α)− g0i
∣∣ > 0)
= N supi∈1,...,N
sup(θ,α)∈Nη
Pr(gi (θ, α) 6= g0i
).
Moreover, the proof of Lemma A4 shows that there exists a non-stochastic bT such that, for η such that
(A8) is satisfied:
supi∈1,...,N
sup(θ,α)∈Nη
Pr(gi (θ, α) 6= g0i
)≤ bT = o(T−δ).
Hence we have, for all δ > 0:
sup(θ,α)∈Nη
Pr
(sup
i∈1,...,N
∣∣gi (θ, α)− g0i∣∣ > 0
)≤ NbT = o
(NT−δ
).
This implies (16), and completes the proof of Theorem 2.
A.3 Proof of Corollary 1
We have:
√NT
(θ − θ0
)=
(1
NT
N∑
i=1
T∑
t=1
(xit − xg0
i t
)(xit − xg0
i t
)′)−1(
1√NT
N∑
i=1
T∑
t=1
(xit − xg0
i t
)vit
),
which tends to N(0,Σ−1
θ ΩθΣ−1θ
)by Assumption 3.a-3.c and the Cramer theorem. Result (20) then follows
from the fact that√NT
(θ − θ
)= op(1) and the Mann-Wald lemma.
Next we have, for all (g, t):
αgt =
∑Ni=1 1
g0i = g
(yit − x′itθ
)
∑Ni=1 1 g0i = g
= α0gt +
(∑Ni=1 1
g0i = g
xit∑N
i=1 1 g0i = g
)′ (θ0 − θ
)+
∑Ni=1 1
g0i = g
vit∑N
i=1 1 g0i = g.
Now, using Assumptions 1.b and 2.a as well as the above we have:
(∑Ni=1 1
g0i = g
xit∑N
i=1 1 g0i = g
)′ (θ0 − θ
)= op
(1√N
).
Hence:
√N(αgt − α0
gt
)=
1√N
∑Ni=1 1
g0i = g
vit
1N
∑Ni=1 1 g0i = g
+ op(1),
and (21) follows from a similar argument as before.
This ends the proof of Corollary 1.
58
A.4 Proof of Proposition 1
Let θ = plimN→∞ θ, and αg = plimN→∞ αg for g ∈ 1, 2, where the probability limits are taken for fixed T
as N tends to infinity. We assume without loss of generality that α1 ≤ α2.
Following the arguments in Pollard (1981), it can be shown that the pseudo-true values θ and αg satisfy:
E
[T∑
t=1
xit(vit + x′it
(θ0 − θ
))+
T∑
t=1
xit1
vi ≤ x′i
(θ − θ0
)+α1 + α2
2− α0
(α0 − α1
)
+T∑
t=1
xit1
vi > x′i
(θ − θ0
)+α1 + α2
2− α0
(α0 − α2
)]
= 0, (A16)
E
[1
vi ≤ x′i
(θ − θ0
)+α1 + α2
2− α0
(vi + x′i
(θ0 − θ
)+ α0 − α1
)]= 0, (A17)
E
[1
vi > x′i
(θ − θ0
)+α1 + α2
2− α0
(vi + x′i
(θ0 − θ
)+ α0 − α2
)]= 0. (A18)
Now, let a1 and a2 be the solutions of:
TE
[1
vi ≤
a1 + a22
− α0
(vi + α0 − a1
)]= 0, (A19)
TE
[1
vi >
a1 + a22
− α0
(vi + α0 − a2
)]= 0. (A20)
We note that(θ0, a1, a2
)satisfies the moment restrictions (A16)-(A18) because, as vitt and xitt are
independent of each other we have:
E
[T∑
t=1
xitvit +
T∑
t=1
xit1
vi ≤
a1 + a22
− α0
(α0 − a1
)+
T∑
t=1
xit1
vi >
a1 + a22
− α0
(α0 − a2
)]
= 0 + E
[T∑
t=1
xit
]E
[1
vi ≤
a1 + a22
− α0
(α0 − a1
)+ 1
vi >
a1 + a22
− α0
(α0 − a2
)]
︸ ︷︷ ︸=0
,
where we have used that the sum of the left-hand sides in (A19) and (A20) is zero.
Provided the solution to the population moment restrictions (A16)-(A18) be unique,42 it thus follows that:
(θ, α1, α2
)=(θ0, a1, a2
). (A21)
Hence θp→ θ0. In addition, it follows from (A19)-(A20) and (A21) that:
E
[1
vi ≤
α1 + α2
2− α0
(vi + α0 − α1
)]= 0,
E
[1
vi >
α1 + α2
2− α0
(vi + α0 − α2
)]= 0.
In particular we have, by symmetry: (α1 + α2)/2 = α0. So:
α1 = α0 − E (vi| vi ≤ 0) ,
42Uniqueness of the population minimum is a key ingredient for showing that(θ, α
)p→(θ, α
)as N tends to infinity
(Pollard, 1981).
59
and likewise for α2. The final result comes from the normality assumption, as:
E (vi| vi ≤ 0) = − σ√T
φ(0)
Φ(0)= −σ
√2
πT.
This ends the proof of Proposition 1.
B Complements
B.1 A finite mixture interpretation
In this section, we note that the grouped fixed-effects estimator may be interpreted as maximizing the (pseudo)
likelihood of a finite mixture model. Making the link with finite mixtures is insightful, as finite mixture modelling
is widely used in economic and statistical applications.
We shall conduct the discussion in the case of the linear model (1), although the equivalence applies to
nonlinear models also. To state the equivalence result, let σ > 0 be a scaling parameter. Then, it is easy to see
that the GFE estimator of (θ, α) satisfies:
(θ, α
)= argmax
(θ,α)∈Θ×AGT
[max
π1,...,πN
N∑
i=1
ln
(G∑
g=1
πig
1
(2πσ2)T2
exp
(− 1
2σ2
T∑
t=1
(yit − x′itθ − αgt)2
))],
(B22)
where the maximum is taken over all probability vectors πi in the unit simplex of RG. The result comes from
the fact that the individual-specific πi are unrestricted in (B22).43 Note also that the identity (B22) holds for
any choice of σ.
Identity (B22) shows that the GFE estimator may be interpreted as the maximizer of the pseudo-likelihood
of a mixture-of-normals model, where the mixing probabilities are individual-specific and unrestricted. This
contrasts with standard finite mixture modelling (McLachlan and Peel, 2000), which typically specifies the
group probabilities πg(xi) as functions of the covariates. In comparison, in the grouped fixed-effects approach
the group probabilities πgi = πg(i) are unrestricted functions of the individual dummies.
B.2 Adding prior information
To proceed, suppose that the a priori information takes the form of prior probabilities on group membership,
πig denoting the prior probability that unit i belongs to group g. A penalized GFE estimator of (θ, α) is:
(θ(π), α(π)
)= argmin
(θ,α)∈Θ×AGT
N∑
i=1
T∑
t=1
(yit − x′itθ − α
g(πi)
i (θ,α)t
)2, (B23)
43Specifically, given (θ, α) values the maximum is achieved at:
πi (θ, α) = argmaxπi
G∑
g=1
πig
1
(2πσ2)T2
exp
(
−1
2σ2
T∑
t=1
(yit − x
′
itθ − αgt
)2)
,
yielding:
πig(θ, α) = 1 gi(θ, α) = g , for all g.
60
where the estimated group variables are now:
g(πi)i (θ, α) = argmin
g∈1,...,G
T∑
t=1
(yit − x′itθ − αgt)2 − C lnπig, (B24)
and where C > 0 is a penalty term. The penalty specifies the respective weights that prior and data information
have in estimation.44
Note that computation of the penalized GFE estimator is very similar to that of the GFE estimator given
by (4).45 In addition, the penalized and unpenalized GFE estimators are asymptotically equivalent under the
conditions given in Section 4, provided prior information be non-dogmatic in the following sense.
Assumption B1 (prior probabilities) The prior probabilities are non-dogmatic is the sense that, for some
ε > 0:
ε < πig < 1− ε, for all (i, g).
We have the following result.
Corollary 2 (penalized GFE) Let the assumptions of Corollary 1 hold, and let π = πig be a set of prior
probabilities that satisfies Assumption B1. Then we have, asymptotically:
√NT
(θ(π) − θ0
)d→ N (0, Vθ) . (B25)
Proof. The proof closely follows that of Theorem 2 and Corollary 1. The key difference with the proof of
Theorem 2 appears in the proof of Lemma A4. It is useful to define the following quantity:
Z(π)ig (θ, α) = 1g0i 6= g1
T∑
t=1
(yit − x′itθ − αgt)2 − C lnπig ≤
T∑
t=1
(yit − x′itθ − αg0
i t
)2− C lnπig0
i
.
Then instead of bounding Zig(θ, α) the proof consists in bounding Z(π)ig (θ, α), the only difference being the
following extra term in AT :
A4T = |−C lnπig + C lnπig| ,
which is bounded as follows:
A4T ≤ C ln
(1− ε
ε
),
where we have used Assumption B1.
B.3 Grouped fixed-effects in unbalanced panels
In this section of the appendix we consider an unbalanced panel whose maximum time length is T . We denote
as dit the indicator variable that takes value one if observations yit and xit belong to the dataset, zero if not.
44A possible choice, motivated by the special case of the normal linear model, is C = 2σ2, where σ2 = E(v2it). In
practice, one may approximate σ2 by taking the mean of (OLS) squared residuals.45We checked in numerical experiments that adding prior information tends to alleviate the local minima problem
documented in Section 3, although it does not fully solve it.
61
We adopt the convention that dityit = 0 and ditxit = 0 when the latter situation happens. Lastly, it is assumed
that xit and vit are (weakly) uncorrelated given dit.
The GFE estimator is then:
(θ, α, γ
)= argmin
(θ,α,γ)∈Θ×ANT×ΓG
N∑
i=1
T∑
t=1
dit (yit − x′itθ − αgit)2. (B26)
Turning to computation, one difference with Algorithm 1 arises in the update step, as it may happen that:
ngt = #i ∈ 1, ..., N, g(s+1)
i = g, dit = 1
is zero, for some (g, t) ∈ 1, ..., G × 1, ..., T. In this case there are no observations to compute α(s+1)gt and
the algorithm stops. When using Algorithm 2, we start a local search (i.e., Step 5) as soon as ngt = 0 for (g, t)
some value.
C Additional results
C.1 Monte Carlo exercise
We start by comparing the estimation results of Table 3– some of which show sizable bias– with estimates
obtained using a natural alternative: the interactive fixed-effects estimator. For the simulated dataset with
G = 3, we estimate the interactive FE estimator of Bai (2009) allowing for three factors. Even though this
estimator, like GFE, is consistent as N and T tend to infinity, the results of 1, 000 Monte Carlo replications
show very substantial biases: the mean of the autoregressive parameter and the coefficient of xit are −.356 and
.155, respectively. These results suggest that the more parsimonious GFE estimator may dominate interactive
FE in relatively short panels.46
As for group-specific time effects, Figure C1 shows the pointwise means of αgt across 1, 000 simulations.
When G = 3, all three time profiles are shifted downwards relative to the true ones by a similar amount. When
G = 5 the bias affects the various groups in slightly different ways. The overall patterns of heterogeneity are
well reproduced on average.47
The evidence in Section 5 is based on a design with i.i.d. normal errors, which might seem too favorable
given that the asymptotic behavior of the GFE estimator crucially depends on tail and dependence properties
of errors. To address this concern, we report results using a different DGP, in which errors are resampled (with
replacement) from the unit-specific vectors of GFE residuals. Note that, given the nature of the original data,
these residuals exhibit serial correlation and are clearly not normally distributed.48 Tables C1 and C2 report
the mean and standard deviation of the GFE estimator for θ across 1, 000 simulations. Compared with the
46Bai (2009) discusses bias reduction in interactive FE models with strictly exogenous regressors. Moon and Weidner
(2010a) provide truncation-based bias reduction formulas in models with predetermined regressors. Note that, in contrast
with interactive FE, the GFE estimator is automatically (higher-order) bias-reducing, even in the presence of lagged
outcomes or general predetermined regressors.47We also have computed the finite-sample variances of the group-specific time effects, and compared them with the
clustered estimator (22). As in Table 4, the results show some sizable differences between the two.48The dependent variable– the Freedom House indicator of democracy, one of the two measures that we use in the
empirical application– has actually only 7 points of support.
62
i.i.d. normal case, the results show small-sample biases of a similar order, but stronger underestimation of the
finite-sample variance.49
As a last exercise, we check the performance of the BIC criterion (27) to estimate the number of groups.
To do so, we count the number of times that BIC selects a given G, across 100 simulated datasets. The results
reported in Table C3 suggest that the criterion performs reasonably well, even in cases where the true number
of groups is relatively large (G0 = 10).50 In addition, we also run simulations where the number of groups G
used in estimation differs from the true number G0. Figure C2 shows that the mean and standard deviation of
the GFE estimator of common parameters are little affected when G > G0 and G0 = 3, consistently with the
discussion in Section 5, although we do observe some increase in the finite-sample dispersion of the estimator
as G grows.
C.2 Tables and figures, Monte Carlo exercise
Table C1: Bias of the GFE estimator (alternative DGP)
θ1 (coeff. yi,t−1) θ2 (coeff. xit)θ2
1−θ1Misclassified
True GFE True GFE True GFE
G = 3 .407 .381 .089 .099 .151 .163 9.47%
G = 5 .255 .314 .079 .082 .107 .125 8.71%
G = 10 .277 .322 .075 .074 .104 .109 25.24%
Note: See the notes to Table 3. Unit-specific sequences of errors are drawn with replacement from the estimated
GFE residuals.
Table C2: Standard deviation of the GFE estimator (alternative DGP)
θ1 (coeff. yi,t−1) θ2 (coeff. xit)θ2
1−θ1
Asymptotic Monte Carlo Asymptotic Monte Carlo Asymptotic Monte Carlo
G = 3 .049 .118 .0104 .0162 .012 .028
G = 5 .042 .125 .0084 .0103 .011 .033
G = 10 .039 .064 .0067 .0086 .009 .013
Note: See the notes to Tables 4. Unit-specific sequences of errors are drawn with replacement from the estimated
GFE residuals.
49The results for group-specific time effects are very similar to those shown in Figure C1 and are omitted.50We also tried the alternative choice σ2 G(T+N−G)
NTln(NT ) for the penalty, instead of σ2 GT+N+K
NTln(NT ) in equation
(27). This corresponds to a common choice of penalty in factor models (e.g., Bai and Ng, 2002). We found that, in this
case, BIC selected 1 group in all 100 simulations, when the truth was G0 = 3. In comparison, Table C3 shows that our
more conservative choice (27) yielded superior results on these data.
63
Table C3: Choice of the number of groups: BIC criterion
G0 = 3
G = 1 2 3 4 5 6
%(G = G) 0 0 98 2 0 0
G0 = 10
G = 7 8 9 10 11 12
%(G = G) 0 10 42 42 6 0
Note: See the notes to Table 3. The results show the number of times that the BIC criterion selected G groups,
when the true number is G0 = 3 (upper panel) or G0 = 10 (lower panel), respectively, out of 100 simulations.
Figure C1: Monte Carlo bias on group-specific time effects
G = 3 G = 5
1 2 3 4 5 6 7−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
1 2 3 4 5 6 7−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
Note: Solid line shows the true values α0gt, dashed lines show the mean of αgt across 1, 000 simulations
with i.i.d. normal errors. x-axis shows time t ∈ 1, ..., 7.
64
Figure C2: GFE, G0 = 3, G 6= G0
θ1 (coeff. yi,t−1) θ2 (coeff. xit)θ2
1−θ1
1 2 3 4 5 6 70
0.2
0.4
0.6
0.8
1
number of groups G (G0 = 3)1 2 3 4 5 6 7
0
0.05
0.1
0.15
number of groups G (G0 = 3)1 2 3 4 5 6 7
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
number of groups G (G0 = 3)
Note: See the notes to Table 3. The DGP has G0 = 3 groups. GFE estimates are computed using G
groups, where G is reported on the x-axis. Solid thick lines and dashed lines indicate the mean and 5%
pointwise confidence bands, respectively, across 1000 simulations. The horizontal solid lines indicate
true parameter values.
65
C.3 Tables and figures, empirical application
Table C4: Income and democracy, OLS and FE
Unbalanced panel Balanced panel
(1) (2) (3) (4)
Lag democracy (θ1) .706(.035)
.379(.051)
.665(.049)
.283(.058)
Lag income (θ2) .072(.010)
.010(.035)
.083(.014)
−.031(.049)
Cumulative income ( θ21−θ1
) .246(.031)
.017(.056)
.246(.019)
−.044(.069)
Observations 945 945 630 630
Countries 150 150 90 90
R-squared .725 .796 .721 .799
Time dummies yes yes yes yes
Country fixed effects no yes no yes
Note: Balanced (1970-2000) and unbalanced (1960-2000) five-year panel data from Acemoglu et al. (2008).
Freedom House indicator of democracy. Robust standard errors clustered at the country level in parentheses.
66
Table C5: Income and democracy, GFE estimates
G Objective BIC Democracy Income Cumulative income
(θ1) (θ2) ( θ21−θ1
)
1 24.301 .052 .665(.049)
.083(.014)
.247(.018)
2 19.847 .046 .601(.041)
.061(.011)
.152(.021)
3 16.599 .042 .407(.052)
.089(.011)
.151(.013)
4 14.319 .039 .302(.054)
.082(.010)
.118(.011)
5 12.593 .037 .255(.050)
.079(.010)
.107(.009)
6 11.132 .036 .465(.043)
.064(.007)
.119(.011)
7 10.059 .035 .403(.043)
.065(.008)
.108(.011)
8 9.251 .035 .333(.044)
.070(.008)
.104(.010)
9 8.426 .034 .312(.045)
.069(.008)
.101(.010)
10∗ 7.749 .034 .277(.049)
.075(.008)
.104(.009)
11 7.218 .034 .293(.042)
.073(.008)
.104(.009)
12 6.809 .034 .304(.044)
.074(.008)
.107(.010)
13 6.391 .035 .236(.040)
.072(.009)
.094(.009)
14 5.996 .035 .237(.042)
.071(.009)
.094(.009)
15 5.664 .035 .244(.043)
.071(.009)
.094(.009)
FE 17.517 − .284(.058)
−.031(.049)
−.044(.069)
Note: See the notes to Figure 1. The table reports the value of the objective function, the Bayesian
information criterion, and coefficient estimates with their standard errors for the GFE estimates with
various values for the number of groups G. The parameter σ2 in BIC was computed using Gmax = 15.
The last row in the table shows the same figures for the fixed-effects model.
67
Table C6: Income and democracy, GFE estimates with country-specific FE
G Objective Democracy Income Cumulative income
(θ1) (θ2) ( θ21−θ1
)
1 17.517 .284(.058)
−.031(.049)
−.044(.069)
2 12.859 .061(.049)
−.038(.027)
−.040(.029)
3 10.400 −.033(.043)
−.035(.027)
−.034(.027)
4 9.221 −.072(.046)
.045(.027)
.042(.025)
5 8.174 −.093(.042)
−.013(.026)
−.011(.024)
Note: See the notes to Table C5. The table reports GFE estimates in deviations to country-specific
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1207 Juan José Ganuza and Fernando Gomez: “Optional law for firms and
consumers: An economic analysis of opting into the Common European Sales Law”.
1208 Stéphane Bonhomme and Elena Manresa: “Grouped patterns of heterogeneity
